Author: Смыкалова Е.В.
Tags: геометрия задачи по геометрии 7 класс
ISBN: 978-5-7704-0281-0
Year: 2017
Text
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ISBN 978-5-7704-0281-0
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4
CaMOCTOHTeJIbHaH pa60Ta N 1
Bapuanm 1
1. npoBe MTe npSlMYIO H OTMeTbTe Ha HeR 4 TOqKH. CKOJIbKO
OTpe3KOB nOJIyqMJIOCb?
2. Ha npSlMoH o'rMeTbTe nOCJIe OBaTeJIbHO TOqKH M, N, P, 8, T.
CKOJIbKO o'rpeaKOB, Ha KOTOpbIX He JIemHT TOqKa 8?
3. OTMe'rh1'e 'rot(KH A, B, C, D HE TaK, qTo6bI TOqKH A, B, C, D
JIe}K8JIM 1J8 O HOH npHMoH, a TOqKa E He JIemaJIa Ha HeH.
1.lepea Knm bIe Be TOqKH npOBe HTe npHMYIO. CKOJIbKO
Beero I1pHMbIX?
4. MO}J{HO JIM HaqepTHTb Be rrepeceKalOIII;HecSi npSlMbIe H
pnenOJIOmHTb Ha HHX Ba He nepeceKalOIII;HXCH oTpe3Ka?
5. CKOJlbKO TOqeK H8,lJ;O BaRTb Ha npHMoH Mem y TOqKaMH A Ii
B, 'IT06bI BMeCTe C OTpe3KOM AB nOJIyqliJIOCb eCHTb
PU3JIHqHbIX oTpe3KoB?
6. ,l];8HbI qeTbIpe TOqKH, HHKaKHe TPH H3 HHX He JIemaT Ha
O,lJ.HOH npHMOH. MomHO JIH npOBeCTH meCTb npHMbIX TaK,
'IT06bI Kam aH H3 3THX npHMbIX npOXO HJIa pOBHO qepe3
lJ.8e 3a aHHbIe TOqKH?
7. MorYT JIH meCTb npHMbIX HMeTb nHTb TOqeK nepeCeqeHHH?
BapuaKm 2
1. IIpoBe HTe npHMYIO H OTMeTbTe Ha HeH 5 TOqeK. CKOJIbKO
OTpe3KOB nOJIyqHJIOCb?
2. H8 npHMoH OTMeTbTe nOCJIe OBaTeJIbHO TOqKH A, B, C, D.
( KOJIbKO OTpe3KOB, Ha KOTOpbIX He JIemHT TOqKa C?
a. OTMeTbTe TOqKIi L, M, N, P, 8 H T TaK, qTo6bI TOqKH L, M,
N, P, S JIemaJIH Ha O HOH npSlMoH, a TOqKa T He JIemaJIa Ha
HeM. Qepe3 Kam bIe Be TOqKH npoBe HTe npHMYIO. CKOJIb-
KO Bcero npHMbIx?
4. MomHo JIH HaqepTHTb Be nepeceKalOIII;HeCH npHMbIe H
paCI10JIOmHTb Ha HHX Ba nepeceKalOIII;HXCH oTpe3Ka?
5. CKOJlbKO TOqeK H8,lJ;O B3HTb Ha npHMoH Mem y TOqKaMH C H
D, tlT06bI BMeCTe C OTpe3KOM CD nOJIyqHJIOCb meCTb pa3-
JIHqHbIX oTpe3KoB?
6. aHbI rrSlTb TOqeK, HliKaKHe TPH H3 HHX He JIemaT Ha O HOH
npSlMOM. MomHo JIH npOBeCTH eCHTb rrpHMbIX TaK, qTo6bI
Kam aSl H3 3THX npHMbIX npOXO HJIa pOBHO qepe3 Be
3a aHHbIe 'rOqKH?
7. MorYT JIH ceMb npHMbIX HMeTb eBHTb TOqeK nepeCeqeHHH?
CaMOCTOHTeJIbHaH pa60Ta N2 1
5
Bapuanm 3
1. IIpoBe HTe npHMYIO H OTMeTbTe Ha HeM 6 TOqeK. CKOJIbKO
OTpe3KOB nOJIyqHJIOCb?
2. Ha npHMoM OTMeTbTe nOCJIe OBaTeJIbHO TOqKH M, N, P, S,
T. CKOJIbKO OTpe3KOB, Ha KOTOpbIX He JIemHT TOqKa P?
3. OTMeTbTe TOqKH A, B, C, D Ii E TaK, qTo6bI TOqKH B, C, D H
E JIemaJIH Ha O HOM npHMoll, a TOqKa A He JIemaJIa Ha HeM.
Qepe3 Kam bIe Be TOqKH npoBe HTe npHMYIO. CKOJIbKO
Bcero npHMbIx?
4. MomHo JIH HaqepTHTb TPH npHMbIe, nepeceKalOIII;HeCH B o -
HOH TOqKe, H paCnOJIOmHTb Ha HHX TPH He nepeceKalOIII;HX-
CH oTpe3Ka?
5. CKOJIbKO TOqeK Ha o B3HTb Ha npHMoR Mem y TOqKaMH M H
N, qTo6bI BMeCTe C OTpe3KOM MN nOJIyqHJIOCb eCHTb pa3-
JIHqHbIX oTpe3KoB?
6. ,1];aHbI qeTbIpe TOqKH, HHK8KHe TPH H3 HHX He JIemaT Ha
O HOM npHMOM. IIpoBecTH npHMbIe TaK, qTo6bI Kam aH H3
3THX npHMbIX npOXO HJI8 pOBHO qepe3 Be 3a aHHbIe
TOqKH. CKOJIbKO 6y eT np.HMbIx?
7. MorYT JIH nHTb npHMbIX HMeTb qeTbIpe TOqKH nepeCeqeHHH?
Bapuanm 4
1. IIpoBe HTe npHMYIO H OTMeTbTe Ha HeR 7 TOqeK. CKOJIbKO
OTpe3KOB nOJIyqHJIOCb?
2. Ha npHMoM OTMeTbTe nOCJIe OBaTeJIbHO TOqKH A, B, C, D.
CKOJIbKO OTpe3KOB, Ha KOTOpbIX He JIemHT TOqKa B?
3. OTMeTbTe TOqKH L, M, N, P, S H TTaK, QTo6bI TOqKH M, N,
P, S H T JIemaJIH Ha O HOH npHMoM, a TOqKa L He JIemaJIa Ha
HeM. Qepe3 Kam bIe Be TOqKH npoBe HTe npHMYIO. CKOJIb-
KO Bcero npHMbIx?
4. MomHo JIH HaqepTHTb TPH npHMbIe, nepeceKalOIII;HeCH B o -
HOH TOqKe, H paCnOJIOmHTb Ha HHX TPH nepeceKalOIII;HXCH
oTpe3Ka?
5. CKOJIbKO TOqeK Ha o B3HTb Ha npHMoH Mem y TOqKaMH S H
T, qTo6bI BMeCTe C OTpe3KOM ST nOJIyqHJIOCb meCTb pa3-
JIHqHbIX oTpe3KoB?
6. ,1];aHbI nHTb TOqeK, HHKaKHe TPH H3 HHX He JIemaT Ha O HOM
npHMOM. IIpoBecTH npHMbIe TaK, qTo6bI Kam aH H3 3THX
npHMbIX npOXO HJIa pOBHO qepe3 Be 3a aHHbIe TOqKH.
CKOJIbKO 6y eT npHMbIx?
7. MorYT JIH meCTb npHMbIX HMeTb ceMb TOqeK nepeCeqeHHH?
6
CaMOCTOHTeJIbHaH pa60Ta N2 2
Bapuanm 1
1. IIpoBe MTe npHMYIO H OTMeTbTe Ha HeH 5 TOqeK. CKOJIbKO
OTpe3K08 IIOJlYtIHJlOCb?
2. Ha npHMoH OTMeTbTe nOCJIe,ZJ;OBaTeJIbHO TOqKH M, N, P, 8,
T, R. ( I(OJlbKO OTpe3KOB C KOHD;OM B TOqKe 8?
:i. O'rMe'rh're 'rotIKH A, B, C, D H E TaK, qTo6bI TOqKH A, B, C
JIejJ(HJIH IIn O HOH npHMoH, a TOqKH D H E He JIemaJIH Ha
HeM. tleJ)( a Kam bIe Be TOqKH npoBe HTe npHMYIO. CKOJIb-
I() Heel'O IIpHMbIX?
4. MomHo JIM Ha ,ZJ;BYX nepeCeKalOIIJ;HXCH npHMbIX paCnOJIOmHTb
'leThlpe OTpe3Ka TaR, qTo6bI OHH He HMeJIH 06 HX TOqeK?
5. MOI'y'r JII1 nHTb npHMbIX HMeTb 1 TOqKY nepeCeqeHHH?
6. MOjf(I10 J111 npOBeCTH 11 npHMbIX qepe3 nHTb TOqeK TaK,
'l'I'OUbl IIpHMaH npOXO,ZJ;HJIa He MeHee, qeM qepe3 2 TOqKH?
7. Jloe'I'poI1Tb MBC. OTMeTHTb Be TOqKH D H E BHYTPH, a
'I'O'II(Y f' - BHe TpeyrOJIbHHKa TaK, qTo6bI HHKaKHe TPH TOq-
KH He JlemaJIH Ha O HOH npHMOH. IIocTpOHTb WEF. CKOJIb-
1(0 "cero oTpe3KoB? Ha30BHTe HaH60JIbllIee B03MomHoe
llHCJIO OTpe3KOB.
Bapuanm 2
1. npoBe HTe npHMYIO H OTMeTbTe Ha HeH 6 TOqeK. CKOJIbKO
(YI'pe3KOB nOJIytlHJIOCb?
2. 110 npHMoH OTMeTbTe nOCJIe OBaTeJIbHO TOqKH A, B, C, D,
E. ( KOJIbKO OTpe3KOB C KOHD;OM B TOqKe D?
:i. ()1'MeTbTe TOqKH M, N, P, 8, T H R TaK, qTo6bI TOqKH M, N,
P, S JIemaJIH Ha O HOH npHMoH, a TOqKH T H R He JIemaJIH
HM nen. Qepe3 Kam bIe Be TOqKH npOBe HTe npHMYIO.
( I(OJlbKO Bcero npHMbIx?
4. MomHo J1H Ha BYX nepeceK8lO HXCH npHMbIX paCnOJIOmHTb
Tpl1 OTpe3Ka TaR, qTo6bI OHH HMeJIH O,ZJ;HY o6 yH> TOqKY?
5. Mory'r JIM 'leTbIpe npHMbIe He HMeTb HH O,ZJ;HOH TOqKH nepe-
cetleHMH'!
6. MomHo JJI1 npOBeCTH 16 npHMbIX qepe3 llIeCTb TOqeK TaK,
tITo6bI npHM8H npOXO HJIa He MeHee, qeM qepe3 2 TOqKH?
7. I1ocTpoH'rb CDE. OTMeTHTb TOqKY F BHYTPH TpeyrOJIbHH-
Ka, a Be TOtIKH L H M - BHe TpeyrOJIbHHKa TaK, qTo6bI HH-
KaKHe TPM TOtIKH He JIemaJIH Ha O HOH npHMOH. IIocTpOHTb
f1FLM. CKOJlbKO Bcero oTpe3KoB? Ha30BHTe HaH60JIbllIee
B03MomHoe tIHCJIO OTpe3KOB.
CaMOCTOHTeJIbHaH pa60Ta N2 2
7
Bapuaum 3
1. IIpoBe,ZJ;HTe npHMYIO H OTMeTbTe Ha HeR 7 TOqeK. CKOJIbKO
OTpe3KOB nOJIyqHJIOCb?
2. Ha npHMoR OTMeTbTe nOCJIe,ZJ;OBaTeJIbHO TOqKIi M, N, P, S,
T, R. CKOJIbKO OTpe3KOB C KOH OM B TOqKe P?
3. OTMeTbTe TOqKH A, B, C, D HE TaK, qTo6bI TOqKH C, D Ii E
JIemaJIH Ha O,ZJ;HOR npHMOR, a TOqKH A H B He JIemaJIH Ha
HeR. Qepe3 Kam,ZJ;bIe ,ZJ;Be TOqKH npoBe,ZJ;HTe npHMYIO. CKOJIb-
KO Bcero npHMbIx?
4. MomHo JIH Ha ,ZJ;BYX nepeceKalO HXCH npHMbIX paCnOJIOmHTb
nHTb OTpe3KOB TaK, qTo6bI OHH He HMeJIH 06 HX TOqeK?
5. MorYT JIH meCTb npHMbIX HMeTb 1 TOqKY nepeCeqeHHH?
6. MomHO JIH npOBeCTH 4 npHMbIe qepea TPH TOqKH TaK,
qTo6bI npHMaH npOXO,ZJ;HJIa He MeHee, qeM qepe3 2 TOqKH?
7. IIocTpoHTb LlABC. OTMeTHTb ,ZJ;Be TOqKH PHS BHYTPH, a
TOqKY T - BHe TpeyrOJIbHHKa TaK, qTo6bI HHKaKHe TPH TOq-
KH He JIemaJIH Ha O,ZJ;HOR npHMOH. IIocTpOHTb MST. CKOJIb-
KO Bcero oTpe3KoB? Ha30BHTe HaH60JIbmee B03MomHoe
qHCJIO OTpe3KOB.
Bapuaum 4
1. IIpoBe,ZJ;HTe npHMYIO H OTMeTbTe Ha HeR 8 TOqeK. CKOJIbKO
OTpe3KOB nOJIyqHJIOCb?
2. Ha npHMoR OTMeTbTe nOCJIe,ZJ;OBaTeJIbHO TOqKH A, B, C, D,
E. CKOJIbKO OTpe3KOB C KOH OM B TOqKe B?
3. OTMeTbTe TOqKH M N, P, S, T H R TaK, qTo6bI TOqKH P, S,
T H R JIemaJIH Ha O,ZJ;HOR npHMoH, a TOqKH M H N He JIemaJIH
Ha HeR. Qepe3 Kam,ZJ;bIe ,ZJ;Be TOqKH npoBe,ZJ;HTe npHMYIO.
CKOJIbKO Bcero npHMbIx?
4. MomHo JIH Ha ,ZJ;BYX nepeCeKalOIIJ;HXCH npHMbIX paCnOJIOmHTb
qeTbIpe OTpe3Ka TaR, qTo6bI OHH HMeJIH O,ZJ;HY o6m;ylO TOqKY?
5. MorYT JIH ceMb npHMbIX He HMeTb HH O,ZJ;HOR TOqKH nepece-
qeHHH?
6. MomHO JIH npOBeCTH 7 npHMbIX qepe3 qeTbIpe TOqKH TaK,
qTo6bI npHMaH npOXO,ZJ;HJIa He MeHee, qeM qepe3 2 TOqKH?
7. IIocTpoHTb t1CDE. OTMeTHTb TOqKY A BHYTPH TpeyrOJIbHH-
Ka, a ,ZJ;Be TOqKH B H C - BHe TpeyrOJIbHHKa TaK, qTo6bI HM-
KaKHe TPH TOqKIi He JIemaJIH Ha O,ZJ;HOH npHMOR. IIocTpOMTb
MBC. CKOJIbKO Bcero oTpe3KoB? Ha30BHTe HaH60JIblllee
B03MomHoe qHCJIO OTpeaKOB.
8
CaMOCTOHTeJIbHaH pa60Ta N2 3
Bapuanm 1
1. HaqepTHTe Jlyq AB. MOmHO JIM rrpOBeCTM Ba JIyqa C Ha-
qaJIOM B TOqKe C TaK, qT06bI O MH H3 JIyqeii rrepeCeKaJI
6bI JIyq AB, a pyroii He rrepeCeKaJI 6bI ero?
2. OTMeTbTe Ha npHMoii 3 TOqKM. CKOJIbKO rrOJIyqMJIOCb JIY-
qeH'1
3. TO IKa A - HaqanO Tpex JIyqeii. MomHO JIM rrpOBeCTM OT-
pe30K BC TaK, qTo6bI OH rrepeCeKaJI TOJIbKO O HH JIyq M3
Tpex?
4. Ha4epTHTe 4 rrpHMbIe, rrepeceKalOID;MeCH Bce B o Hoii
TO IKe o. HaqepTMTe Ba JIyqa C HaqaJIOM B TOqKe O.
CKOJIbKO Bcero JIyqeii?
5. HaqepTHTe TPM rrorrapHO rrepeceKalOID;MeCH rrpHMbIe C
1'peMH TOqKaMM rrepeCeqeHMH. CKOJIbKO Bcero JIyqeii?
6. HaqepTHTe rrpHMoyrOJIbHMK ABCD. IIpoBe MTe rrpH-
MbIe AC M BD, 0 - TOqKa rrepeCeqeHMH 3TMX rrpHMbIX.
CKOJlbKO Bcero Jlyqeii?
7. MomHO JIM M3 100 OTpe3KOB rrOCTpOMTb JIyq
Bapuanm 2
1. HaqepTHTe JIyq BC. MomHO JIM rrpOBeCTH TPM JIyqa C Ha-
qaJIOM B TOqKe A TaK, qTo6bI OHM He rrepeCeKaJIM JIyq
BC?
2. OTMeTbTe Ha rrpHMoii 2 TOqKH. CKOJIbKO rrOJIyqMJIOCb JIY-
qeH?
3. TOqK8 B - HaqaJIO qeTbIpex JIyqeii. MomHO JIM rrpOBeCTH
OTpeaOK CD TaK, qTo6bI OH rrepeCeKaJI TOJIbKO Ba JIytIa M3
qeTbIpex?
4. HaqepTHTe 3 rrpHMbIe, rrepeceKalOID;MeCH Bce B O HOH
TOqKe A. HaqepTMTe Ba JIyqa C HaqaJIOM B TOqKe A.
CKOJIbKO Bcero JIyqeii?
5. HaqepTHTe rrpHMoyrOJIbHMK ABCD. IIpoBe MTe rrpH-
Mble AB, BC, CD, DA. CKOJIbKO Bcero JIyqeii?
6. HaqepTHTe KBa paT KLMN. npoBe HTe rrpHMbIe KM H
LN, 0 - TOqKa rrepeCeqeHHH 3THX rrpHMbIX. CKOJIbKO
Bcero JlyqeH?
7. MomHo JIH H3 50 OTpe3KOB rrOCTpOHTb JIyq?
CaMOCTOHTeJIbHaH pa60Ta M 3
9
Bapuanm 3
1. Ha"tlepTHTe JIY"tI AB. MomHo JIH rrpoBecTH ,n;Ba JIY"tIa C
Ha"tlaJIOM B TO"tlKe M TaK, "tITo6bI O,n;HH H3 JIY"tIeH rrepece-
KaJI 6bI JIY"tI AB, a ,n;PyroH He rrepeCeKaJI 6bI ero?
2. OTMeTbTe Ha rrpHMoH 5 TO"tleK. CKOJIbKO rrOJIY"tIHJIOCb
JIY"tIeH?
3. TO"tlKa A - Ha"tlaJIO rrHTH JIY"tIeH. MomHo JIH rrpOBeCTH
OTpe30K BC TaK, "tITo6bI OH rrepeCeK8JI TOJIbKO ,n;Ba JI)T"tIa H3
rrHTH?
4. Ha"tlepTHTe 4 rrpHMbIe, rrepeceKalO HeCH Bce B O,n;HOH
TO"tlKe O. Ha"tlepTHTe TPH JIY"tIa C Ha"tlaJIOM B TO"tlKe O.
CKOJIbKO Bcero JIY"tIeH?
5. Ha"tlepTHTe TpeyrOJIbHHK MNP. IIpoBe,n;HTe rrpHMbIe
MN, NP, PM. CKOJIbKO Bcero JIY"tIeH?
6. Ha"tlepTHTe rrpHMoyrOJIbHHK ABCD. IIpoBe,n;HTe rrpH-
MYIO AC. CKOJIbKO Bcero JIY"tIeH?
7. MomHo JIH H3 200 OTpe3KOB rrOCTpOHTb JI)T"tI?
Bapuanm 4
1. Ha"tlepTHTe JIY"tI BC. MomHo JIH rrpoBecTH TPH JIY"tIa C Ha-
"tIaJIOM B TO"tlKe A TaK, "tITo6bI OHH rrepeCeKaJIH 6bI JIY"tI
BC?
2. OTMeTbTe Ha rrpHMoH 4 TO"tlKH. CKOJIbKO rrOJIY"tIHJIOCb
JIY"tIeH?
3. TO"tlKa B - Ha"tl8JIO rneCTH JI)T"tIeH. MomHo JIH rrpOBeCTH
OTpe30K CD TaR, "tITo6bI OH rrepeCeK8JI TOJIbKO TPH JI)T"tIa H3
rnecTH?
4. Ha"tlepTHTe 3 rrpHMbIe, rrepeceKalO HeCH Bce B O,n;HOH
TO"tlKe A. Ha"tlepTHTe "tIeTbIpe JI)T"tIa C Ha"tlaJIOM B TO"tlKe A.
CKOJIbKO Bcero JIY"tIeH?
5. Ha"tlepTHTe rrpHMoyrOJIbHHK CDEF. IIpoBe,n;HTe rrpH-
MbIe CD, DE, EF, FC. CKOJIbKO Bcero JIY"tIeH?
6. Ha"tlepTHTe KBa,n;paT KLMN. IIpoBe,n;HTe rrpHMYIO LN.
CKOJIbKO Bcero JIY"tIeH?
7. MomHo JIH 113 150 OTpe3KOB rrOCTpOHTb JIyq?
10
CaMOCTOSlTeJIbHaSi pa60Ta N2 4
Bapuaum 1
1. TpH npSlMbIe nepeceKalOTCSI B O HOH TOqKe. CKOJIbKO
yrJIOB, MeHbIIIHX paaBepHYToro, 06paayeTcSI?
2. HaqepTHTe MBC. OTMeTbTe TOqKY D BO BHYTpeHHeH 06JIaCTH
TpeyrOJIbHHKa. MO:>KHO JIH npoBeCTH JIyq C HaqaJIOM B TOqKe D
TaK, qT06bI OH nepeCeKaJI JIyq AB H He nepeCeKaJI JIyq BC?
3. HaqepTMTe OCTpbIH LMNP. MOjRHO JIH npOBeCTH JIyqH C
HaqaJIOM B TOqKe N TaK, qT06bI nOJIyqHJIOCb IIIeCTb yrJIOB,
MeHbIIIHX paaBepHYToro?
4. HaqepTHTe Be npSlMbIe AB H CD, nepeceKalOrn;HecSi B
TOqKe O. IIpoBe HTe JIyq OF BHYTPH LAOC. CKOJIbKO
yrJIOB, MeHbllIHX paaBepHYToro?
5. CKOJIbKO npSlMbIX Ha o npOBeCTH qepea aa aHHYIO TOqKY,
qT06bI npH HX nepeCeqeHHH 06paaOBaJIOCb 24 yrJIa, MeHb-
IIIHX paaBepHYToro?
6. YrJIbI AOB, BOC, COD, DOE, EOA HMelOT 06rn;ylO BepIIIH-
HY O. MO:>KHO JIH npOBeCTH npSlMYIO, He npoxo Slrn;ylO qepea
TOqKY 0 TaK, qT06bI OHa nepeCeKaJIa TPH JIyqa?
7. Be napbI napaJIJIeJIbHbIX npSlMbIX rrepeceKalOTCSI. CKOJIb-
KO 06paayeTcSI yrJIOB, MeHbIIIHX paaBepHYToro?
Bapuaum 2
1. CKOJIbKO yrJIOB, MeHbIIIHX paaBepHYToro, 06paayeTcSI npH
nepeCeqeHHH BYX npSlMbIx?
2. HaqepTHTe l1 DEF. OrMeTbTe TOqKY K BO BHYTpeHHeH 06JIaCTH
TpeyroJIbHHKa. MO:>KHO JIH npoBeCTH JIyq C HaqaJIOM B TOqKe K
TaR, qT06bI OH nepeceKaJI JIyq EF H He nepeceKaJI JIyq DE?
3. HaqepTHTe TynoH LABC. MO:>KHO JIH npOBeCTH JIyqH C Ha-
qaJIOM B TOqKe B TaK, qT06bI nOJIyqHJIOCb eCSlTb yrJIOB,
MeHbIIIHX paaBepHYToro?
4. HaqepTHTe Be npSlMbIe CD H EF, nepeceKalOrn;HecSi B TOq-
Ke O. IIpoBe HTe JIyqH OA H OB BHYTPH LCOE. CKOJIbKO yr-
JIOB, MeHbllIHX paaBepHYToro?
5. CKOJIbKO npSlMbIX Ha,zJ;O npOBeCTH qepea aa aHHYIO TOqKY,
qTo6bI npH HX nepeCeqeHHH 06paaOBaJIOCb 12 yrJIOB, MeHb-
IIIHX paaBepHYToro?
6. YrJIbI KOL, LOM, MON, NOP, POK HMelOT 06rn;ylO Bep-
IIIHHY O. MO:>KHO JIH npOBeCTH npSlMYIO, He npoxo Slrn;ylO
qepea TOqKY 0 TaK, qT06bI OHa nepeCeKaJIa qeTbIpe JIyqa?
7. IIpSiMaSi nepeceKaeT TPH napaJIJIeJIbHbIe npSlMbIe. CKOJIbKO
06paayeTcSI yrJIOB, MeHbIIIHX paaBepHYToro?
CaMOCTOHTeJIbHaH pa60Ta N 4
11
Bapuanm 3
1. TpH rrpHMbIe rrepeceKalOTCH B OtJ;HOH TOqKe. CKOJIbKO
yrJIOB, MeHbllIHX paaBepHYToro, o6paayeTcH?
2. HaqepTHTe !!1 ABC. OTMeTbTe TOqKY E BO BHYTpeHHeH o6JIac-
TH TpeyrOJIbHHKa. MO:iKHO JIH rrpoBeCTH .rryq C HaqarrOM B TOqKe
E TaK, qTo6bI OH rrepeCeK8JI JIytI BC H He rrepeCeKaJI JIytI AB?
3. HaqepTHTe OCTpbIH LMPS. MO:iKHO JIH rrpOBeCTH JIyqH C Ha-
qaJIOM B TOqKe P TaK, QTo6bI rrOJIyqHJIOCb llIeCTb yrJIOB,
MeHbllIHX paaBepHYToro?
4. HaqepTHTe ,n;Be rrpHMbIe AB H CD, rrepeceKalOm;HeCH B
TOqKe O. IIpoBe,n;HTe JIyq ON BHYTPH LAOD. CKOJIbKO
yrJIOB, MeHbllIHX paaBepHYToro?
5. CKOJIbKO rrpHMbIX Ha,n;o npOBeCTH qepea aa,n;aHHYIO TOqKY,
qTo6bI rrpH HX rrepeCeqeHHH nOJIyqHJIOCb 24 yrJIa, MeHbllIHX
paaBepHYToro?
6. YrJIbI AOB, BOC, COD, DOE, EOA HMelOT o6m;ylO BepllIH-
HY O. MO:iKHO JIH rrpOBeCTH rrpHMYIO, He rrpOXO,n;Hm;ylO qepea
TOqKY 0 TaK, qTo6bI OHa nepeCeKaJIa ,n;Ba JIyqa?
7. IIpHMaH rrepeceKaeT qeTbIpe rrapaJIJIeJIbHbIe rrpHMbIe.
CKOJIbKO o6paayeTcH yrJIOB, MeHbllIHX paaBepHYToro?
Bapuanm 4
1. CKOJIbKO yrJIOB, MeHbllIHX paaBepHYToro, o6paayeTcH rrpH
rrepeCeqeHHH ,n;BYX rrpHMbIx?
2. HaqepTHTe!!1 DEF. OTMeTbTe TOqKY M BO BHYTpeHHeH o6JIac-
TH TpeyrOJIbHHKa. MO:iKHO JIH rrpoBeCTH .rryq C HaqarrOM B TOqKe
M TaK, qTo6bI OH rrepeCeKaJI JIyq DE H He rrepeCeKaJI JIytI EF?
3. HaqepTHTe TyrroH LBCD. MO:iKHO JIH rrpOBeCTH JIyqH C
HaqaJIOM B TOqKe C TaK, qTo6bI rrOJIyqHJIOCb ,n;eCHTb yrJIOB,
MeHbllIHX paaBepHYToro?
4. HaqepTHTe ,n;Be rrpHMbIe CD H EF, rrepeceKalOm;HeCH B
TOqKe O. IIpoBe,n;HTe JIyqH OA H OB BHYTPH LCOF. CKOJIb-
KO yrJIOB, MeHbllIHX paaBepHYToro?
5. CKOJIbKO rrpHMbIX Ha,n;o rrpOBeCTH qepea aa,n;aHHYIO TOqKY,
qTo6bI rrpH HX rrepeCeqeHHH rrOJIyqHJIOCb 12 yr JIOB, MeHb-
llIHX pa3BepHYToro?
6. YrJIbI KOL, LOM, MON, NOP, POK HMelOT o6m;ylO Bep-
llIHHY o. MO:iKHO JIH rrpOBeCTH rrpHMYIO, He rrpOXO,n;Hm;ylO
qepea TOqKY 0 TaK, qTo6bI OHa rrepeCeKaJIa TPH JIyqa?
7. IIpHMaH rrepeceKaeT rrHTb rrapaJIJIeJIbHbIX rrpHMbIX. CKOJIb-
KO o6paayeTcH yrJIOB, MeHbllIHX paaBepHYToro?
12
CaMOCTOHTeJIbHaH pa60Ta N2 5
Bapuanm 1
1. IIocTpoHTe pa3BepHYTbIH LAOB, LAOC = 60° II LBOD = 60°.
TOqKH C H D B O HOH nOJIynJIOCKOCTH. CKOJIbKO 06paaOB8JI0Cb
nap paBHbIX yrJIOB, MeHbillHX paaBepHyroro?
2. HaqepTHTe ABC. TOqKa D BHYTPH ABC. MomHO JIH npoBecTH
JIyq CD, eJIHm;HH ABC Ha Ba paBHbIX TpeyrOJIbHHKa?
3. HaqepTHTe TYIIOH LABC. CKOJIbKO JIyqeH C Haq8JI0M B TOqKe B
BHYTPH LABC HMO npOBeCTH, qT06bI npH 3TOM 06paaOB8JI0Cb 10
nap paBHbIX yrJIOB, MeHbIIIHX paaBepHyroro?
4. BoceMb npHMbIX, npoBe eHHbIx qepea O HY TOqKY eJIHT nJIOC-
KOCTb Ha 16 paBHbIX qaCTeH. KaKOBa rpMYCHaH Mepa HaHMeHb-
llIHX OCTpbIX yrJIOB?
5. HaqePTHTe ABC. OrMeTbTe TOqKY D BO BHellIHeH 06JIacTH Tpe-
yroJIhHHKa. MomHO Jrn rrpoBecru .rryq C Haq8JI0M B TOq}{e D TaR, qrrQ-
6bI OH nepeceK8JI npHMYJO AB H He nepeceK8JI npHMbIe AC II BC?
6. IIepHMeTp TpeyrOJIbHHKa paBeH 11 CM. O Ha CTopOHa B 2 paaa
MeHbllIe pyroH H Ha 3CM MeHbllIe TpeTbeH. HaH HTe MeHbllIYIO
CTOPOHY TpeyrOJIbHHKa.
7. IIo.nynepHMerrp npHMoyroJIbHHKa paBeH 13 CM, a O Ha Ha 3THX CTO-
poH 60JIbllIe pyroH Ha 3 CM. H HTe n.rrO:W;Mb npHMoyroJIbHHKa.
Bapuanm 2
1. IIocTpoHTe paaBepHYTbIH LAOB, LAOC = 45°, LCOD = 45° H
LBOE = 45°. TOqKH C, D H E B O HOH rrOJIynJIOCKOCTH. CKOJIbKO
06paaOB8JI0Cb nap paBHbIX yrJIOB, MeHbllIHX pa3BepHYToro?
2. HaqepTHTe BCD. TOqKa E BHe BCD. MomHo JIH npoBeCTH JIyq
DE, eJIH:W;HH BCD Ha Ba paBHbIX TpeyrOJIbHHKa?
3. HaqepTHTe OCTpbIH LMNP. CKOJIbKO JIyqeH C Haq8JI0M B TOqKe
N BHyrpH LMNP HMO npOBeCTH, qT06bI npH 3TOM 06paaOB8JI0Cb
qeTbIpe napbI paBHbIX yrJIOB, MeHbllIHX paaBepHYToro?
4. qeTbIpe rrpHMbIX, npoBe eHHbIx qepea O HY TOqKY eJIHT nJIOC-
KOCTb Ha 8 paBHbIX qaCTeH. KaKOBa rpMYCHaH Mepa HaHMeHb-
III II X OCTpbIX yr JIOB?
5. HatleprrHTe ABC. OrMeTbTe TOtlKY D BO BHeIIIHeH 06JIacTH Tpe-
yroJIbHHKa. MOmHO JIH rrpoBecTH .rryq C Haq8JI0M B TOq}{e D TaR, qrrQ-
6bI OH nepeceK8JI np.HMbIe AB II BC II He nepeceK8JI npHMYlO AC?
6. IIepHMeTp TpeyrOJIbHHKa paBeH 16 CM. O Ha CTopOHa B 2 paaa
60JIbIIIe pyroH H Ha 1CM MeHbllIe TpeTbeH. HaH HTe 6oJIbIIIYlO
CTOPOHY TpeyrOJIbHHKa.
7. IIoJIyrrepHMeTp npHMoyrOJIbHHKa paBeH 10 CM, a O,lJ;Ha Ha 3THX
CTOpOH MeHbllIe pyroH Ha 2 CM. HaH,lJ;HTe rrJIo:w;a b rrpHMO-
yrOJIbHHKa.
CaMOCTOHTeJIbHaH pa60Ta N2 5
13
Bapuanm 3
1. IIocTpoHTe paaBepHYTbIH LMON, LMOP = 60° H LNOS =
= 60°. TOqKH PHS B OAHOH nOJIynJIOCKOCTH. CKOJIbKO 06paaOBa-
JIOCb nap paBHbIX yrJIOB, MeHbmHX paaBepHYToro?
2. HaqepTHTe CDE. TOqKa F BHyTpH CDE. MomHo JIH npOBec-
TH JIyq EF, AeJIH HH CDE Ha ABa paBHbIX TpeyrOJIbHHKa?
3. HaqepTHTe TynoH LAOB. CKOJIbKO JIyqeH C HaqaJIOM B TOqKe 0
BHYTPH LAOB HaAo npOBeCTH, qT06bI npH aTOM 06paaOBaJIOCb
10 nap paBHbIx yrJIOB, MeHbmHX paaBepHYToro?
4. ,lJ;ecHTb npHMbIX, npoBeAeHHbIx qepea OAHY TOqKY AeJIHT nJIOC-
KOCTb Ha 20 paBHbIX qaCTeH. KaKoBa rpaAYCHaH Mepa HaHMeHb-
mHX OCTpbIX yrJIOB?
5. HaqepTHTe CDE. OrMeTbTe TOqKY F BO BHemHeH 06JIacTH Tpe-
yroJIbHHKa. MomHo JIH npoBecTH .rryq C Haq8JI0M B TOqKe F TaR, qTO-
6bIOH nepeceK8JI IIpHMYJO CD H He nepeceK8JI npHMbIe DE H CE?
6. IIepHMeTp TpeyrOJIbHHKa paBeH 28 CM. OAHa CTopOHa B 2 paaa
MeHbme APyroH H Ha 8 CM MeHbme TpeTbeH. HaHAHTe MeHbmylO
CTOPOHY TpeyrOJIbHHKa.
7. IIoJIYIIepHMeTp npHMoyrOJIbHHKa paBeH 11 CM, a OAHa Ha aTHX CTO-
poH 60JIbme ApyrOH Ha 3 CM. HaiiAHTe nJIO Mb npHMoyrOJIbHHKa.
Bapuanm 4
1. IIocTpoHTe paaBepHYTbIH LMON, LMOP = 45°, LPOS = 45° H
LNOT = 45°. TOqKH P, S H T B OAHOH nOJIynJIOCKOCTH. CKOJIbKO
06paaOBaJIOCb nap paBHbIX yrJIOB, MeHbmHX paaBepHYToro?
2. HaqepTHTe DEF. TOqKa K BHe DEF. MomHo JIH npOBeCTH
JIyq FK, AeJIH HH DEF Ha ABa paBHbIX TpeyrOJIbHHKa?
3. HaqepTHTe OCTpbIH LMON. CKOJIbKO JIyqeH C HaqaJIOM B TOqKe
o BHYTPH LMON HMO npOBeCTH, qT06bI npH aTOM 06paaoBa-
JIOCb qeTbIpe napbI paBHbIx yrJIOB, MeHbmHX paaBepHYToro?
4. IIHTb npHMbIX, npoBeAeHHbIx qepea OAHY TOqKY AeJIHT nJIOC-
KOCTb Ha 10 paBHbIX qaCTeH. KaKOBa rpMYCHaH Mepa HaHMeHb-
mHX OCTpbIX yrJIOB?
5. HaqepTHTe ABC. OTMeTbTe TOqKY M BO BHemHeH 06JIaCTH
TpeyrOJIbHHKa. MomHo JIH npOBeCTH JIyq C HaqaJIOM B TOqKe M
TaK, qT06bI OH nepeCeKaJI npHMbIe AC H BC H He nepeCeKaJI
npHMYIO AB?
6. IIepHMeTp TpeyrOJIbHHKa paBeH 22 CM. OAHa CTopOHa B 2 paaa
60JIbme APyroH H Ha 2 CM MeHbme TpeTbeH. HaHAHTe 60JIbmylO
CTOPOHY TpeyrOJIbHHKa.
7. IIoJIyIIepHMerrp IIpHMOyroJIbHHKa paBeH 9 CM, a OAHa Ha 3THX CTO-
poH MeHbme ApyroH Ha 3 CM. H HTe IIJIO Mb IIpHMOyroJIbHHKa.
14
CaMOCTOSlTeJlbHaSi pa60Ta N2 6
Bapuanm 1
1. Ha Jlyqe C HaqanOM 0 OTMe"tJ:eHbI TOqKM M, N M STaR, tlTO TOtl-
Ka S JlemHT Mem y TOtlKaMM N M M, a TOtlKa M - MemAY TOtl-
KaMM 0 H S. KaROH OTpeaOK HaH60JlbllIMH OM, OS MJlH ON?
2. Ha npHMoH OTMetleHbI TOqKM A, B, C MD. TOtlKa C JlemMT
Mem,lJ;Y A H B TaK, tlTO AB = CD. MomeT JIM Cepe,lJ;MHa
OTpe3Ka AD 6bITb Cepe,lJ;MHOH OTpe3Ka BC?
3. Ha npSlMoH n OT TOqKM A OTJlOmeHbI ,lJ;Ba OTpe3Ka TaK, tlTO
AC 60JlbllIe, tleM AB, M TOtlKa A JlemMT Mem,lJ;Y TOtlKaMM B M
C. OT TOtlKM C OTJlOmeH OTpe30K CM TaK, tlTO BM = AC,
TOqKa M Mem,lJ;Y A M C. CpaBHMTe OTpe3KM MC M AB.
4. OM - 6McceKTpMca LAOB. JIYtI OC BHYTPM LAOM, a Jly-q
OD - BHYTPM LMOB, LAOC = LBOD. HBJlSleTCSI JlH OM
6McceKTpMcoH LAOD?
5. OB - Jly-q BHYTPM LAOC. MomHO JIM rrpOBeCTM JlYtI OE TaK,
tlTo6bI LAOC = LBOE?
6. B TpeyrOJlbHMKe rrepBbIH yroJl B 2 pa3a 60JlbllIe BToporo M Ha
20 0 MeHbllIe TpeTbero. HaH,lJ;MTe 60JlbllIMH yroJl.
7. CYMMa ,lJ;BYX yr JlOB TpeyrOJlbHMKa paBHa 80°, a O,lJ;MH M3 3TMX
yr JlOB 60JlbllIe ,n;pyroro Ha 60°. HaH,lJ;MTe MeHbllIMH yroJl
TpeyroJlbHMKa.
Bapuanm 2
1. Ha.rryqe C HatlanOM 0 OTMetleHbI TOtlKM A, B M C TaR, tlTO TOtlKa
C JlemMT MemAY TOtlKaMM A M B, a TOqKa A - Mem,ny TOtlKaMM
o H C. KaROM OTpe30K HaH60JlbllIMH OA, OB MJlM OC?
2. Ha rrpHMoH OTMetleHbI TOtlKM B, C, D M E. TOtlKa D JlemMT
Mem,n;y C M B TaK, tlTO BC = DE. MomeT JIM Cepe,lJ;MHa
OTpe3Ka BE 6bITb cepe,n;MHoH OTpe3Ka CD?
3. Ha rrpSiMoH n OT TOtlKM A OTJlOmeHbI ,lJ;Ba OTpe3Ka TaK, tlTO
AC 60JlbllIe, tleM AB, M TOtlKa A JlemHT Mem,lJ;y TOtlKaMM B M
C. OT TOtlKM C OTJlOmeH OTpe30K CN TaK, tlTO BA = NC,
TOtlKa N Mem,lJ;y A M C. CpaBHMTe OTpe3KH BN M AC.
4.0P - 6McceKTpMca LBOC. JIy-q OD BHYTPM LBOP, a JlYtl
OE - BHYTPH L POC, LBOD = LCOE. HBJlSleTCSI JIM OP
6McceKTpHCOH LDOC?
5. OC - Jlytl BHYTPH LAOB. MomHo JIM rrpOBeCTH Jly-q OD TaK,
tlTo6bI LAOC = LBOD?
6. B TpeyroJlbHHKe rrepBbIH yroJl B 3 paaa 60JlbllIe BToporo M Ha
30° 60JlbllIe TpeTbero. HaH,lJ;HTe MeHbllIMH yroJl.
7. CYMMa ,lJ;BYX yr JlOB TpeyroJlbHMKa paBHa 120 0 , a O,n;MH M3
3TMX yr JlOB MeHbllIe .n;pyroro Ha 20°. HaM,lJ;HTe 60JlbllIMH
yroJl TpeyroJlbHMKa.
CaMOCTOSlTeJIbHaSi pa60Ta N2 6
15
Bapuanm 3
1. Ha JI e C HaqarrOM 0 OTMeqeHbI TQqKM M, N M 8 TaR, qTO TQq-
Ka 8 JIemMT Memmr TOqKaMM N M M, a TOqKa M - Memmr TOqKa-
MIl 0 M 8. KaROM OTpe30K HaHMeHbIlIMM OM, 08 MJIM ON?
2. Ha IIpHMOM OTMeqeHbI TOqKM A, B, C M D. TOqKa C JIemMT
MemtJ;y A M B TaK, qTO AB = CD. MomeT JIM cepetJ;MHa
OTpe3Ka AD 6bITb cepetJ;HHoM OTpe3Ka CD?
3. Ha IIpSiMOM n OT TOqKM A OTJIOmeHbI tJ;Ba OTpe3Ka TaK, qTO
AP MeHbllIe, qeM AC, M TOqKa A JIemMT MemtJ;y TOqKaMM P M
C. OT TOqKM C OTJIOmeH OTpe30K CN TaK, qTO PN = AC,
TOqKa N MemtJ;y A Me. CpaBHMTe OTpe3KM NC M AP.
4. OM - 6McceKTpMca LAOB. JIyq OC BHYTPM LAOM, a JIyq
OD - BHYTPM LMOB, LAOC = LBOD. HBJISleTCSI JIM OM
6McceKTpMcoM LCOD?
5. ON - JIyq BHYTPM LCOD. MomHo JIM IIpOBeCTM JIyq OM
TaK, qTo6bI LCOM = LNOD?
6. B TpeyrOJIbHHKe IIepBbIM yrOJI B 2 pa3a 60JIbllIe BToporo M Ha
20° 60JIbllIe TpeTbero. HaMtJ;MTe 60JIbllIMM yrOJI.
7. CYMMa tJ;BYX yr JIOB TpeyrOJIbHMKa paBHa 90° , a OtJ;MH M3 3TMX
yr JIOB 60JIbllIe tJ;pyroro Ha 40°. HaMtJ;MTe MeHbllIMM yrOJI
TpeyrOJIbHMKa.
Bapuanm 4
1. Ha.rryqe C HaqaJIOM 0 OTMeqeHbI TOqKM A, B M C TaR, qTO TOqKa
C JIemMT Memmr TOqKaMM A M B, a TOqKa A - Memmr TOqKaMM
o M C. KaROM OTpe30K HaHMeHbllIMM OA, OB MJIM OC?
2. Ha IIpSiMOM OTMeqeHbI TOqKM B, C, D M E. TOqKa D JIemMT
MemtJ;y C M B TaK, qTO BC = DE. MomeT JIM cepetJ;MHa OTpe3-
Ka BE 6bITb cepetJ;MHoM OTpe3Ka BC?
3. Ha IIpSiMOM n OT TOqKH A OTJIOmeHbI tJ;Ba OTpe3Ka TaK, qTO
AB MeHbllIe, qeM AK, M TOqKa A JIemMT MemtJ;y TOqKaMM B
M K. OT TOqKM K OTJIOmeH OTpe30K KM TaK, qTO AB = KM,
TOqKa M MemtJ;y A M K. CpaBHMTe OTpe3KM BM M AK.
4. OP - 6McceKTpMca LBOC. JI OD BHYTPM LBOP, a JI OE -
BHYTPM LPOC, LBOD = LCOE. HBJISleTCSI JIM OP 6McceK-
TpMCOM LDOE?
5. OK - JIyq BHYTPM LDOE. MomHo JIM IIpOBeCTM JIyq OL TaK,
qTo6bI LDOK = LLOE?
6. B TpeyrOJIbHMKe IIepBbIM yrOJI B 3 paaa 60JIbllIe BToporo M Ha
40° MeHbllIe TpeTbero. HaHtJ;MTe MeHbllIMH yrOJI.
7. CYMMa tJ;BYX yr JIOB TpeyrOJIbHMKa paBHa 110°, a OtJ;MH M3
3TMX yrJIOB MeHbllIe tJ;pyroro Ha 70°. HaHtJ;MTe 60JIbllIMH
yrOJI TpeyrOJIbHMKa.
16
CaMOCTOHTeJIbHaH pa60Ta N2 7
Bapuanm 1
1. TOqKIi A, B Ii C JIemaT Ha OtJ;HOH npHMOH. AB = 8 CM, CB = 9 CM.
KaKoM MomeT 6bITb tJ;JIHHa OTpe3Ka AC?
2. JIemaT JIM TOqKIi A, B H C Ha OtJ;HOH rrpHMoH, eCJIH AC = 6 CM,
AB =- 2 CM, BC = 5 CM?
3. Ha npHMoM OTMeqeHbI TOqKIi 0, A Ii B TaK, qTO OA = 14 CM,
OB - 8 CM. HaHtJ;IiTe paCCTOHHlie MemtJ;y cepetJ;IiHaMIi
OTpe3KOB OA Ii OB, eCJIIi TOqKa 0 He JIemliT Ha OTpe3Ke AB.
4. O'l'pe30K, paBHbIH 34 CM, pa3,lJ;eJIeH Ha TPH HepaBHbIX OTpe3-
Ka. PaccToHHHe Mem,l);y cepetJ;IiHaMH KpaHHIiX OTpe3KOB 18 CM.
HaMtJ;IiTe tJ;JIHHY cpetJ;Hero OTpe3Ka.
5. O'rpe30K tJ;JIIiHbI x pa3tJ;eJIeH Ha qeTbIpe paBHbIe qaCTIi. HaH-
,lJ;MTe paCCTOHHlie MemtJ;y cepetJ;IiHaMIi KpaHHlix qaCTeH.
6. TOqKIi A, B H C JIemaT Ha OtJ;HOH rrpHMoH, TOqKH 0 H P - ce-
petJ;IiHbI OTpe3KOB AB Ii AC. BepHo JIIi paBeHCTBO
BC = 2.0P?
7. OTMeTbTe rrHTb TOqeK TaK, qTo6bI HIiKaKHe TPIi He JIemaJIli
Ha OtJ;HOH rrpHMOH. Qepe3 KamtJ;ylO rrapy TOqeK rrpOBetJ;HTe
npHMYIO. CKOJIbKO rrOJIyqliJIOCb rrpHMbIX?
Bapuanm 2
1. TOqKH B, C H D JIemaT Ha OtJ;HOH rrpHMoH. BC = 7 CM, DC = 9 CM.
KaKoH MomeT 6bITb tJ;JIHHa OTpe3Ka BD?
2. JIemaT JIH TOqKIi K, N H M Ha O,lJ;HOH rrpHMoH, eCJIH KM = 7 CM,
KN = 2 CM, NM = 5 CM?
3. Ha rrpHMoH OTMeqeHbI TOqKIi 0, A Ii B TaK, qTO OA = 12 CM,
OB = 6 CM. HaHtJ;IiTe paCCTOHHHe MemtJ;y cepetJ;IiHaMIi OT-
pe3KOB OA HOB, eCJIH TOqKa 0 JIemHT Ha oTpe3Ke AB.
4. OTpe30K, paBHbIH 32 CM, pa3tJ;eJIeH Ha TPH HepaBHbIX OTpe3-
Ka. PaCCTOHHHe MemAY cepetJ;IiHaMli KpaHHHX OT})e3KOB 20 CM.
HaH,lJ;MTe tJ;JIIiHY cpetJ;Hero OTpe3Ka.
5. OTpe30K tJ;JIIiHbI x pa3tJ;eJIeH Ha TPH paBHbIe qaCTH. HaHtJ;HTe
paCCTOHHlie MemtJ;y cepetJ;HHaMH KpaHHHx qaCTeH.
6. TOqKM K, L Ii M JIemaT Ha OtJ;HOH rrpHMoH, TOqKH 0 H P - ce-
pe,lJ;MHbI OTpe3KOB KL Ii KM. BepHo JIIi paBeHCTBO
LM = 2.0P?
7. OTMeTbTe qeTblpe TOqKH TaK, qTo6bI HIiKaKHe TPIi He
JIemaJIIi Ha otJ;HOM rrpHMoH. Qepe3 KamtJ;ylO rrapy TOqeK
npOBetJ;HTe rrpHMYIO. CKOJIbKO rrOJIyqHJIOCb rrpHMbIx?
CaMOCTOHTeJIbHaH pa60Ta N2 7
17
Bapuaum 3
1. TOqKM A, B M C JIemaT Ha OtJ;HOM IIpHMOH. AB = 7 CM, CB = 8 CM.
KaKoM MomeT 6bITb tJ;JIMHa OTpe3Ka AC?
2. JIemaT JIM TOqKM A, B M C H8 OtJ;HOM rrpHMoM, eCJIM AC = 8 CM,
AB = 3 CM, BC = 6 CM?
3. Ha rrpHMoH OTMeqeHbI TOqKM 0, AM B TaK, qTO OA = 16 CM,
OB = 12 CM. HaHtJ;MTe paCCTOHHMe MemtJ;y cepetJ;MH8MM OT-
pe3KOB OA MOB, eCJIM TOqKa 0 He JIemMT Ha OTpe3Ke AB.
4. OTpe30K, paBHbIH 36 CM, p83,n;eJIeH Ha TPM Hep8BHbIX OTpe3-
Ka. PaCCTOHHMe MemtJ;y cepetJ;MHaMM KpaMHMX OTpe3KOB 26
CM. HaHtJ;MTe tJ;JIMHY cpe,n;Hero OTpe3Ka.
5. OTpe30K tJ;JIMHbI b p83tJ;eJIeH H8 qeTbIpe paBHbIe qaCTM. HaH-
tJ;MTe paCCTOHHMe MemtJ;y cepe,n;MH8MM KpaHHMx qaCTeH.
6. TOqKM B, C M D JIem8T H8 OtJ;HOH npHMoM, TOqKM 0 M M -
cepetJ;MHbI OTpe3KOB BC M BD. BepHO JIM paBeHcTBo
CD = 2. OM?
7. OTMeTbTe rrHTb TOqeK T8K, qTo6bI HMKaKMe TPM He JIemaJIM
Ha OtJ;HOH rrpHMOH. Qepe3 KamtJ;ylO rrapy TOqeK rrpOBetJ;MTe
rrpHMYIO. CKOJIbKO rrOJIyqMJIOCb rrpHMbIX?
Bapuaum 4
1. TOqKM B, C M D JIemaT Ha OtJ;HOM rrpHMoH. BC = 5 CM, DC = 7 CM.
KaKoH MomeT 6bITb tJ;JIMHa OTpe3Ka BD?
2. JIemaT JIM TOqKM K, N M M Ha OtJ;HOH rrpHMOH, eCJIM KM = 9 CM,
KN = 3 CM, NM = 6 CM?
3. Ha rrpHMoH OTMeqeHbI TOqKM 0, AM B TaK, qTO OA = 18 CM,
OB = 4 CM. HaHtJ;MTe paCCTOHHMe MemtJ;y cepetJ;MH8MM OT-
pe3KOB OA MOB, eCJIM TOqK8 0 JIemMT Ha oTpe3Ke AB.
4. OTpe30K, paBHbIH 38 CM, pa3,n;eJIeH H8 TPM HepaBHbIX OTpe3-
Ka. PaCCTOHHMe MemtJ;y cepetJ;MHaMM Kp8HHMX OTpe3KOB
22 CM. HaHtJ;MTe tJ;JIMHY cpetJ;Hero OTpe3Ka.
5. OTpe30K tJ;JIMHbI b pa3,n;eJIeH Ha TPM paBHbIe qaCTM. HaMtJ;MTe
paCCTOHHMe MemtJ;y cepetJ;MHaMM KpaHHMx qaCTeH.
6. TOqKM C, D ME JIemaT Ha OtJ;HOH rrpHMOM, TOqKM 0 M N - ce-
petJ;MHbI OTpe3KOB CD M CE. BepHO JIM paBeHcTBo
DE = 2.0N?
7. OTMeTbTe qeTbIpe TOqKM TaK, qTo6bI HMKaKMe TPM He JIema-
JIM Ha OtJ;HOH rrpHMOH. Qepe3 KamtJ;ylO rrapy TOqeK rrpOBetJ;M-
Te rrpHMYIO. CKOJIbKO rrOJIyqMJIOCb rrpHMbIX?
18
CaMOCTOHTeJIbHaH pa60Ta NQ 8
Bapuanm 1
1. Ha npHMO" OTMctIeHbI TOtlKH 0, A H B TaK, tlTO OA = 16 CM,
OB = 10 eM. HaM,n;HTe paCCTOHHHe Mem,n;y cepe,n;HHaMH
OTpe3KOB OA Ii OB, eCJIH TOtlKa 0 JIemHT Ha OTpe3Ke AB.
2. JIemaT JIH TOtlKH M, NH P Hao,ll;HoM rrpHMoH, eCJIH MP = 7 CM,
MN : CM, NP = 4 CM?
3. OTpeaOI{, ,lI;JlHHa KOToporo paBHa x, pa3,n;eJIeH rrpOH3BOJIb-
1I0H TOlIKOM Ha ,n;Ba OTpe3Ka. HaH,n;HTe paCCTOHHHe Mem,n;y
eepe HHaMH 3THX OTpe3KOB.
4. ()rrpeaoK ,n;JIHHbI x pa3,n;eJIeH Ha BoceMb paBHbIX tlaCTeM. HaH-
1 I1're paCCTOHHHe Mem,n;y cepe,n;HHaMH KpaHHHx tlaCTeH.
5. O'rpe30K 46 CM pa3,n;eJIeH Ha tleTbIpe HepaBHbIX OTpe3Ka.
»aCCTOHHHe Mem,n;y cepe,n;HHaMH KpaMHHx OTpe3KOB paBHO
40 CM. HaM,n;HTe paCCTOHHHe Mem,n;y cepe,n;HHaMH cpe,n;HHX
o'rpe3KOB.
6. IIJlo a,n;b KBa,n;paTa paBHa 25 KB. CM. HaH,n;HTe rrepHMeTp
KBa,n;paTa.
7. aH OTpe30K AB. HaH,n;HTe Ha rrpHMoH AB Bce TaKHe TOtlKH
fJ, ,n;JIH KOTOpbIX DA = 3. DB. CKOJIbKO TaKHX TOtleK?
Bapuanm 2
1. Ha npHMoH OTMetleHbI TOtlKH 0, A H B TaK, tlTO OA = 18 CM,
OB = 8 CM. HaM,lI;HTe paCCTOHHHe Mem,n;y Cepe,ll;HHaMH
01'pe3KOB OA HOB, eCJIH TOtlKa 0 He JIemHT Ha OTpe3Ke AB.
2. JlemaT JIH TOtlKH A, B H C Ha O,n;HOM rrpHMoM, eCJIH AC = 6 CM,
AB = 4 CM, BC = 3 CM?
:J. (}rpe30K, ,n;JIHHa KOToporo paBHa n, pa3,n;eJIeH rrpOH3BOJIb-
HOit TOtlKOM Ha ,lI;Ba OTpe3Ka. HaM,n;HTe paCCTOHHHe Mem,l);y
cepe HHaMH 3THX OTpe3KOB.
4. 01'peaoK ,n;JIHHbI n paa,n;eJIeH Ha meCTb paBHbIX tlaCTeH. HaH-
I1Te paCCTOHHHe Mem,n;y cepe,n;HHaMH KpaHHHx tlaCTeH.
5. OTpcaoK 48 CM pa3,n;eJIeH Ha tleTbIpe HepaBHbIX OTpe3Ka.
PacC1'OHHl1e Mem,n;y Cepe,ll;HHaMII KpaHHHx OTpe3KOB paBHO
38 CM. Hai1:,n;HTe paCCTOHHHe Mem,ll;y cepe,n;HHaMH cpe,n;HHX
OTpe3KOB.
6. IIepl1MeTp KBa,n;paTa 24 CM. HaM,n;HTe rrJIo:rn;a,n;b KBa,n;paTa.
7. ,[(aH OTpeaOK AB. HaM,n;HTe Ha rrpHMoM AB Bce TaKHe TOtlKH
C, ,lI;JIH KOTOpbIX CA = 2 · CB. CKOJIbKO TaKHX TOtleK?
CaMOCTOSlTeJIbHaSi pa60Ta N2 8
19
Bapuanm 3
1. Ha rrpSiMoM OTMeqeHbI TOqKH 0, C H D TaK, qTO OC = 14 CM,
OD = 6 CM. HaH HTe paCCTOSlHHe Mem y cepe HHaMH OT-
pe3KOB OC HOD, eCJIH TOqKa 0 JIemHT Ha OTpe3Ke CD.
2. JIemaT JIH TOqKH M, N H P Ha O HOM npSlMoM, eCJIH MP = 9 CM,
MN = 6 CM, NP = 3 CM?
3. OTpe30K, JIHHa KOToporo paBHa m, pa3 eJIeH npOH3BOJIb-
HOM TOqKOM Ha Ba OTpe3Ka. HaM HTe paCCTOSlHHe Mem y
cepe HHaMH 3THX OTpe3KOB.
4. OTpe30K JIHHbI m pa3 eJIeH Ha nSlTb paBHbIX qaCTeM. HaM-
HTe paCCTOSlHHe Mem y cepe HHaMH KpaHHHx qaCTeH.
5. OTpe30K 44 CM pa3 eJIeH Ha qeTbIpe HepaBHbIX OTpe3Ka.
PaccToSlHHe Mem y cepe HHaMH KpaMHHx OTpe3KOB paBHO
32 CM. HaM HTe paCCTO.HHHe Mem y cepe HHaMH cpe HHX
OTpe3KOB.
6. IIJIorn;a b KBa paTa paBHa 49 KB. CM. HaM HTe nepHMeTp
KBa paTa.
7. ,lJ:aH OTpe30K AB. HaM HTe Ha npSlMoM AB Bce TaKHe TOqKH
D, JISI KOTOpbIX DA = 4. DB. CKOJIbKO TaKHX TOqeK?
Bapuanm 4
1. Ha npSlMoM OTMeqeHbI TOqKH 0, C H D TaK, qTO OC = 12 CM,
OD = 4 CM. HaH HTe paCCTOSlHHe Mem y cepe HHaMH OT-
pe3KOB OC HOD, eCJIH TOqKa 0 He JIemHT Ha OTpe3Ke CD.
2. JIemaT JIH TOqKH A, B H C Ha o HoH npSlMoH, eCJIH AC = 10 CM,
AB = 6 CM, BC = 5 CM?
3. OTpe30K, JIHHa KOToporo paBHa k, pa3 eJIeH npOH3BOJIb-
HOM TOqKOM Ha Ba OTpe3Ka. HaM HTe paCCTOSlHHe Mem y
cepe HHaMH 3THX OTpe3KOB.
4. OTpeaoK ,n;JIHHbI k pa3 eJIeH Ha qeTbIpe paBHbIe qaCTH. HaM-
HTe paCCTOSlHHe Mem y cepe HHaMH KpaHHHx qaCTeH.
5. OTpe30K 42 CM pa3 eJIeH Ha qeTbIpe HepaBHbIX OTpe3Ka.
PaccToSlHHe Mem y cepe HHaMH KpaMHHx OTpe3KOB paBHO
34 CM. HaH,n;HTe paCCTO.HHHe Mem y cepe HHaMH cpe HHX
OTpe3KOB.
6. IIepMMeTp KBa paTa 32 CM. HaH HTe nJIOrn;a b KBa paTa.
7. ,lJ:aH OTpe30K AB. HaM HTe Ha rrpSiMoM AB Bce TaKHe TOqKH
C, JISI KOTOpbIX CA = 3. CB. CKOJIbKO TaKHX TOqeK?
20
CaMOCTOHTeJlbHaH pa60Ta N2 9
Bapuanm 1
1. JIyq OC eJIMT / AOB Ha ,lI;Ba yrJla. HaH,lI;HTe LCOB, eCJlH
LAOB = 84 , a LAOC Ha 12° MeHbllIe, qeM LCOB.
2. LAOB HRJIHfYI H qaCTblO LAOC; LAOC = 105°, a LAOB B 4 pa3a
60JIhIIIC, {leM LBOC. HaH,lI;HTe rpa,ll;YCHYIO Mepy LAOB.
:J. J]y.. fl }IHJIHeTCH 6HcceKTpHcoH Hepa3BepHYToro yrJla hk.
MO)I(PT JIM yroJl hn 6bITb rrpHMbIM?
4. / A()l pacrrOJlOmeH BO BHYTpeHHeH 06J1aCTH LCOD, OE H
()f' oMcceKTpHcbI LCOA H "LBOD COOTBeTCTBeHHO; LCOD
11 / AOB B CYMMe COCTaBJlHIOT 180°. .HBJlHeTCH JIH LEOF
IIIUIMhIM?
5. / AOB npHHMJlemHT BHYTpeHHeH 06J1aCTH LCOD; LCOD =
110"; LAOB = 100°. HaH,lI;HTe yroJl, o6pa30BaHHbIH 6HC-
eel{TpHCaMH LAOC H LBOD, eCJlH Jlyq OB BHYTPH LAOD.
6. JI yq ,lI;eJlHT rrpHMoH yroJl Ha ,lI;Ba yr Jla, TaK, qTO nOJlOBHHa O,ll;-
Boro yrJla paBHa TpeTH ,lI;pyroro. HaH,lI;HTe MeHbllIHH yroJl.
7. IlpHMoH yrOJl ,lI;BYMH .rryqaMH, HCXO,ll;HIIJ;HMH H3 ero BepIlIHHbI,
pa3,l1;eJleH Ha TPH yrJla, O,ll;HH H3 KOTOpbIX paBeH pa3HOCTH ,lI;BYX
PyrHx yrJlOB. HaH,lI;HTe BeJlHqHHY 60JIbllIerO H3 3THX yrJIOB.
Bapuanm 2
1. JIyq ON ,lI;eJlHT LCOD Ha ,lI;Ba yrJla. HaH,lI;HTe LNOD, eCJlH
/ COD = 76", a LCON Ha 14° 60JlbllIe, qeM LJ.lOD.
2. / MON HBJlHeTCHqaCTbIOLPON; LPON=84°, aLPOMB 6 pa3
MeHbllIe, qeM LMON. HaH,lI;HTe rpa,ll;YCHYIO Mepy LMON.
a. JIyq m HBJIHeTCH 6HcceKTpHcoH pa3BepHYToro yrJla hk. Mo-
:>KeT JlH yroJl km 6bITb rrpHMbIM?
4. L AOB pacrrOJlOmeH BO BHYfPeHHeH 06J1aCTII LCOD, ON H OM -
6HcceKTpHcbI LAOC H LEOD COOTBeTCTBeHHO; LCOD H LAOB
B CYMMe COCTaBJIHIOT 180". .HBJlHeTCH JIH LNOM OCTPbIM?
5. / ( OD rrpHHMJlemHT BHYTpeHHeH 06JIaCTH LAOB; LAOB =
= 120"; / COD = 80". HaH,lI;HTe yroJl, o6pa30BaHHbIH 6HcceK-
TpHcaMH AOC H LBOD, eCJlH JIyq OD BHYTPH LCOB.
6. JIyq: eJIMT pa3BepHYTbIH yrOJl Ha ,lI;Ba yrJla, TaK, qTO rrOJlOBHHa
O,ll;HorO yrJI8 paBHa TpeTH ,lI;pyroro. HaH HTe 60JlbllIHH yrOJl.
7. IIpHMoM yroJl ,lI;BYMH JIyqaMH, HCXO,ll;HIIJ;HMH H3 ero BepIlIHHbI,
pa3,l1;eJleH Ha TPH yr Jla, O,ll;HH H3 KOTOpbIX paBeH CYMMe ,lI;BYX
,lI;pyrHx yr JIOB. HaH,lI;HTe BeJlHqHHY 60JlbllIerO H3 3THX yr JlOB.
CaMOCTOHTeJIbHaH pa60Ta N2 9
21
Bapuanm 3
1. JIyq OC ,lJ;eJIMT LAOB Ha ,lJ;Ba yrJIa. HaM,lJ;HTe LCOB, eCJIM
LAOB = 82°, a LAOC Ha 16' MeHbIIle, qeM LCOB.
2. LAOBHBJIHeTcH qaCTblO LAOC; LAOC = 126°, a LAOBB 5 pa3
6oJIbIIle, qeM LBOC. HaiI,lJ;IITe rpa.nyCHYJO Mepy LAOB.
3. JIyq k HBJIHeTCH 6IICCeKTpMcoM Hepa3BepHYToro yrJIa nm.
MomeT JIM yrOJI kn 6bITb rrpHMbIM?
4. LCOD pacrrOJIOmeH BO BHYTpeHHeM 06JIacTM LAOB, OM Ii OP -
6McceKTpMcbI LCOA M LEOD COOTBeTCTBeHHO; LAOB M LCOD
B CYMMe COCTaBJIHIOT 180". .HBJIHeTCH JIM yrOJI LMOP rrpH-
MbIM?
5. LAOB rrpMHa,ZJ;JIemMT BHYTpeHHeiI 06JIaCTM LCOD; LCOD =
= 160°; LAOB = 120'. HaM,lJ;IITe yrOJI, o6pa30BaHHbIM 6MC-
ceKTplicaMM LAOC II LBOD, eCJII1 JIyq OB BHYTpM LAOD.
6. JI ,lJ;eJIMT rrpHMoM yrOJI Ha ,lJ;Ba yrJIa, TaK, qTO rrOJIOBMHa O,lJ;-
Horo yr JIa paBHa TpeTM ,lJ;pyroro. HaM,lJ;MTe 60JIbllIMM yrOJI.
7. IIpHMoM yrOJI ,lJ;BYMH JI aMM, MCXO,lJ;HIIJ;MMMM3 ero BepIIlMHbI,
pa3,lJ;eJIeH Ha TPM yr JIa, O,lJ;MH M3 KOTOpbIX paBeH pa3HOCTM ,lJ;BYX
,lJ;pyrMx yrJIOB. HaM,lJ;MTe BeJIMqMHY 60JIbllIerO M3 3TMX yrJIOB.
Bapuanm 4
1. JIyq ON ,lJ;eJIMT LCOD Ha ,lJ;Ba yrJIa. HaM,lJ;MTe LNOD, eCJIII
LCOD = 78°, a LCON Ha 12° 60JIbllIe, qeM LNOD.
2. LMON HRTIHeTCH qaCTbIO LPON; LPON = 96", a LPOM B 7 pa3
MeHbllIe, qeM LMON. H MTe rpa.nyCHYJO Mepy LMON.
3. JIyq h HBJIHeTCH 6McceKTpMcoM pa3BepHYToro yrJIa kn. Mo-
meT JIM yrOJI hn 6bITb rrpHMbIM?
4. LCOD pacrrOJIOmeH BO BHYTpeHHeH 06JIacTM LAOB, OS MOT -
6McceKTpMCbI LCOA M LBOD cooTBeTcTBeHHo; LAOB M LCOD
B CYMMe COCTaBJIHIOT 180°. .HBJIHeTCH JIM yrOJI LSOT OCTpbIM?
5. LCOD rrpMHa,ZJ;JIemMT BHYTpeHHeM 06JIaCTM LAOB; LAOB =
= 150°; LCOD = 110°. HaM,lJ;MTe yrOJI, o6pa30BaHHbIM 6IICCeK-
TpMcaMM LAOC M LEOD, eCJIII JIyq OD BHyrpM LCOB.
6. JIyq ,lJ;eJIMT pa3BepHyrbIM yrOJI Ha ,lJ;Ba yr JIa, TaR, qTO rrOJIOBMHa
O,lJ;HOrO yrJIa paBHa TpeTM ,lJ;pyroro. Haii,lJ;liTe MeHbllIMM yrOJI.
7. IIpHMoM yrOJI ,lJ;BYMH JI aMM, MCXO,lJ;HIIJ;MMMM3 ero BepIlIMHbI,
pa3,lJ;eJIeH Ha TPM yr JIa, O,lJ;MH M3 KOTOpbIX paBeH CYMMe ,lJ;BYX
,lJ;pyrlIx yrJIOB. HaM,lJ;MTe BeJIMqMHY 60JIbllIerO M3 3TliX yrJIOB.
22
CaMOCTO.HTeJIbHa.H pa60Ta N 10
Bapuaum 1
1. LAOB SIBJIHeTCH tlacTblO AOC; LAoe = 100-, a LAOB B 3 pa3a
II e e Ha 20' 60JIhllIe, qeM yro.n LBOe. HaH)J;MTe LBOC.
2. Jlyq b HBJI.HeTCH 6McceKTpHcoll Tyrroro yrJIa hk. MomeT JiM
yrOJI hb 6hITh TyrrhIM?
3. LAOB npMHMJIemMT BHYTpeHHen 06JIaCTM / eOD; LCOD =
= 120 , a LAOB - B 5 pa3 MeHhllIe. Hall)J;HTe yrOJI, o6pa30-
BaHHhIll6lfcceKTpMCaMI1 / AOe H LBOD, eCJIlf JIyq OB BHYT-
pH: yrJIa LAOD.
4. Pa3BepHYThIH yrOJI pa3)J;eJIeH JIY1:IOM, HCXO,n;RIII;I1M 113 ero Bep-
IlIMHbI, Ha )J;Ba yrJIa, TaKHe, 'lITO rneCTaR qaCTb o,n;Horo yrJIa
paBHa TpeTI1 ,n;pyroro. I!aii,n;I1Te 60JIbIIlMH yrOJI.
5. IIpHMoH yrOJI ,n;BYMH oI"Iy-qaMM, I1CXO,n;.HIIJ;MMI1 M3 ero BeplllM-
HbI, pa3,n;eJIeH Ha Tpl1 yrJl B COOTHOIIIeHMH 1 : 2 : 3. Hall,n;MTe
MeHbllIl1H yrOJI.
6. ,I:{Ba paBHhIX TYIIbIX yrJIa MMelOT o6I..QYlO cTopony, a ,n;Be
,n;pyrMe CTOpOHbI B3aMMHO rrepneH,n;MKY JI.HpHbI. Han,lI;HTe
BeJIH'lIHHY Tyrroro yr J[a.
7. BepHo JIM cJIe,n;yrom;ee BbICKaahJB HMe: CYMMa Tpex OCTpbIX
yrJIOB MO.iKeT 6bI1'b paBRa rrpHMOMY yrJIY?
Bapuanm 2
1. LCOD HBJIHeTCH qaCThlO LCOE; LCOE:--:: 110<, a LCODB 2 pa33 M
eII e Ha 20" 60JIbllle, qeM yrOJI L_DOE. Haii,n;uTe LCOD.
2. JIyq k HBJIHeTCH 61Ir.ceKTpMcoM rrpJlMOrO yrJIa nln. Mallie'}' JIl1
yrOJI kn 6bITb OCTpbIM?
3. LCOD rrpIIHa,n;JIeJKMT BHYTpeHHeH 06JIaCTlf LEOF; L _EOF
= 140 0 , a L COD - B 7 pa3 MeHbllie. HaM)J;MTe yrOJI, o6pa30-
BaHHbIH 6McceRTpMcaMM LEOC H / DOF, eCJIH JIY'll OD BHYT-
pM yrJIa LCOF.
4. I1p.HMOH yrOJI paan:eJIeH ..1JYQOM, IICXO,lIRrn;IIM 113 ero BepIIIII-
Hbl, Ha )J.Ba Yl'JUi, TaKMe, "(ITO llIeCTa.H qaCTb OAHoro yr..TIa paB-
HR Tpt: Tll ) pyroro. Haii,n;MTe MeHbllIHM yrOJI.
5. PaJBepHYThIH yrOJI ,l(BYMH .rryqaMH, HCXO,lI;R MMH 113 ero Bep-
UJJ1HbJ, pa3.n;e.i'IeH H8 rrpn yr JIa B COOTHOIIleHlfll 1 : 2 : 3. Haii-
,l];IITe 60JIbllIllM yrOJI.
6. ,[(Ba paBHbIX TYlIbIX yr JIa I1MeIOT 06II YfO C'J'OpOHY, a ,l(Be
,n;pyrHc ( TOpOnhI o6pa3YloT yrOJI 80". Hail.n;llTe Be.i'Ill1JI1HY
Tyrroro yr.TIa.
7. BepHo JII1 CJIe,n;yroru;ee BhJ( Ka3bIBaHMe: pa3HOCTh I BYX TynT:JIX
Yl'JI0B MOiK8T 6blTb paBHa rrpHMOMY Y l'JlY?
CaMOCTOHTeJIhHaH pa60Ta N 10
23
Bapuanm 3
1. LAOB HBJIHeTCH qaCTblO AOC; L.AOC -= 140 , a LAOB B 3 pa3a
II eru;e Ha 20 60JIbIIle, qeM yrOJI LBOC. HaH.LJ;IITe LBOC.
2. JIyq b flBJIHeTC 6HcceKTpIIcoM OCTporo yrJIa hk. MomeT JIll
yrOJI hb 6bITb TynbIM?
3. LAOB npHHa.LJ;JI€mHT BHYTpeHHeH 06JIaCTH LCOD; LCOD =
= 132', a LAOB - B 6 pa3 MeHblIle. HaH,lI;MTe yrOJI, o6paso-
BaHHbIM 6ncceRTpHcaMH LAOe H LBOD, eCJIH JIyq OB BHYT-
pM Yl'.i'I3 LAOD.
4. Pa3BepHYTbIH yrOJI pa3,neJIeH JIYlIOM, IICXO.n;H lIM II3 ero Bep-
IIlIIHbI, Ha )J;Ba yr JIa, TaRIIe, qTO qeTBepTaH qaCTb O,lI;HOrO yr-
JIa paBHa rrOJIOBHHe tJ:pyroro. HaiI,n;IfTe 60JIbIIJMH yrOJI.
5. IIpflMoH yrOJI )J;BYMfl ..TIY"4aMH, HCXOJJ;Hrn.HMH H3 ero BepllIH-
HbI, paa.n;e.neH Ha TPH yr JIa B COOTHOllIeHHlI 2 : 3 : 4. HaM.n;HTe
MeHbIIllli:i yrOJI.
6. .IJ:Ba paBHbIX Tynblx yr JIa HMelOT o6rn.ylO CTOPOHY, a ,l];Be
,lI;pyrHe CTOpOHbI o6pa3YlOT yrOJI 70". HaiI,lI;MTe BeJIlIqlIHY
Tynoro yrJla.
7. BepHo JIH CJIe,n;YIOIII,ee BblCKa3bIBaHl1e: CYMMa Tpex OCTpbIX
yrJIOB MomeT 6hITb paRII3 pa3BepHYToMY yrJIY?
Bapuanm 4
1. LCOD HBJIHeTCH qaCTblO LCOE; LCOE = 130', a LCOD B 2 pa3a If
ern.e Ha 40' 60JIbIIIe, qeM yrOJI LDOE. Han,l];HTe LCOD.
2. JIyq k HBJIHeTCH 6HcceKTpl1coM Tyrroro yrJIa nm. MomeT JIH
yrOJI km 6bITb OCTphIM?
3. LCOD npHHa.n; TJ:ejKHT BHYTpeHHeM 06JIaCTH LEOF; LEOF =
= 160 0 , a LCOD - B 5 pas MeHbllIe. HaM)J;IlTe yrOJI, o6pa30-
BaHHbIll 611cceKTpIIcaMII / EOC H / DOF, eCJIH JIyq OD BHyrr-
pH yrJIa LCOF.
4. II PHMoM yrOJI pa3JJ;eJIeH JI OM, HCXO.n;HIIJ;HM ll3 ero BepllIII-
HhI, Ha ,l];Ba yrJIa, TaKIIe, qTO qeTBepTaH qaCTb O}J.Horo yrJIa
paBH3 nOJIOBI1He .n;pyroro. HaHl].HTe MeHblllHM yrOJI.
5. PaaBepHYThlH YI'OJI ,n;BYMR .HYQaMI1, I1CXO)J;HIIJ;HMH II3 ero BeIJ-
llIHHbI, pa3.n;e"TIeH Ha Tpn yrJIa B COOTHOllieHIIH 2 : 3 : 4. Ha:i1:-
,lI.IITe 60JIbIIlI1H yrOJI.
6. ,IJ;Ba paBHbIX TYilbIX yr JIa H:'deIOT o6rn.ylO CTOPOHY, a ,r(BC
,n,pyrl1e r.rropOHhI o6pa3YIO'I' yro.J1 60'. H aM,rJ;I1Te Re.TIl1qHHY
Tyrroro yr JIa.
7. RepHo JIM CJIe,n;YIOIIJ;ee BbICKaabIBaHIIe: CYMMa .n;BYX TyrrbJ x
yrJIOB MomeT 6hITb paBHa pa3BepHYTOMY yrJIY?
24
( aMOeTO.HTeJIbHaH pa60Ta N 11
BapUaltnl J
1. I-Iaii HTe IIPpnaBcpHYTbIe yr JIbI, o6pa30BaHHbIe npl1 nepece-
\feUHH / BYX np.HMbIX, eCJII1 CYMMa ,lJ;BYX 113 HI1X paBHa 130°.
2. ( YM MH 'I'P( X yrJIOB, o6pa30BaHHbIx npl1 nepeCeqeHl111 ,lJ;BYX
IIIUIMhlX paHHa 230°. HaH,lJ;I1Te Hal1MeHbllIl1H yrOJI.
:J. II pH IlepeeelJeHHI1 ,lJ;BYX npHMbIX 06pa30BaJII1Cb qeTbIpe yr-
.TU' M(,III.IIJ( pa3BepHYTOrO. HaH,lJ;I1Te 60JIbllII1H yrOJI, 3HaH,
'ITO l'pn/ yeHbIe MepbI ,lJ;BYX 113 HI1X OTHOCHTCH KaK 4 : 5.
4., A()J H / BOC - CMemHbIe, OM - 611cceKTpl1ca LAOB,
.JlY'1 ()N IIpHHa,lJ;JIemI1T BHYTpeHHeH 06JIaCTI1 LBOC 11 nep-
11('1I/ HH:YJUlpeH OM. HBJIHeTCH JII1 ON 611CCeKTpl1COH LBOC?
5. I PJ)lIO JIM YTBepm,lJ;eHl1e: CYMMa CMemHbIX yrJIOB paBHa
1 HO '!
H. l1a 'I'O'IKH 0 npOBe,lJ;eHbI JIyqH OC, OD 11 OE, JIyq OD BHyrplI
/ (YJE, OC 11 OD - nepneH,lJ;I1KYJIHpHbI. YroJI, o6pa30BaHHbIM
fiHCeeKTpl1CaMI1 LCOD 11 LDOE, paBeH 75°. Haii,lJ;I1Te LDOE.
7. IIJUI MOM yroJI pa3,lJ;eJIeH JIyqOM, I1CXO,lJ;HIIJ;I1M 113 ero BepllIl1-
IIhl, Ha ,lJ;Ba yrJIa, TaKl1e, qTO qeTBepTb O,lJ;HOrO yrJIa paBHa
I UITOH qaCTI1 ,lJ;pyroro. HaH,lJ;IITe MeHbllIl1H yrOJI
Bapuanm 2
1. I la H,lJ,MTe Hepa3BepHYTbIe yr JIbI, o6pa30BaHHbIe npl1 nepece-
tJPlIl1M ,lJ;BYX npHMbIX, eCJII1 CYMMa ,lJ;BYX 113 HI1X paBHa 210°.
2. ( YMM8 Tpex yrJIOB, o6pa30BaHHbIx npl1 nepeCeqeHl111 ,lJ;BYX
npHMblX paBHa 240°. HaH,lJ;I1Te Hal16oJIbllII1H yrOJI.
3. IIpl1 lIepeCeqeHl111 ,lJ;BYX npHMbIX o6pa30BaJIHCb qeTbIpe yr-
JIa MeHbllle pa3BepHYToro. HaH,lJ;I1Te MeHbllIl1H yrOJI, 3HaH,
lJ TO I'pa,lJ;YCHbIe MepbI ,lJ;BYX 113 HI1X OTHOCHTCH KaK 2 : 3.
4. A()B II LBOC - CMemHbIe, ON - 611cceKTpl1ca LBOC, JIyq
OP npl1H8 JIeml1T BHYTpeHHeH 06JIaCTI1 LAOB 11 nepneH-
,lJ;HKYJIHpeH ON. HBJIHeTCH JII1 OP 611CCeKTpl1COH LAOB?
5. BepHo JIM YTBepm,lJ;eHl1e: CMemHbIe yrJIbI paBHbI?
6. 113 TOlJKII 0 IIpOBe,lJ;eHbI JIyql1 OC, OD 11 OE, JIyq OD BHyrpl1
L_COE, OC H OD - nepneH,lJ;I1KYJIHpHbI. YroJI, o6pa30BaHHbIM
6HcceKTpMc8MI1 LCOD 11 LDOE, paBeH 65°. HaH,lJ;I1Te LDOE.
7. Pa3BepHYThIH yrOJI pa3,lJ;eJIeH JIyqOM, I1CXO,lJ;HIIJ;I1M 113 ero
BepllIl1Hbl, Ha ,lJ;Ba yrJIa, TaKl1e, qTO qeTBepTb O,lJ;HOrO yrJIa
paBHa IIHTOH qaCTI1 ,lJ;pyroro. HaH,lJ;I1Te 60JIbllII1M yrOJI.
CaMOCTOHTeJIbHaH pa60Ta N2 11
25
Bapuanm 3
1. HaH,ZJ;MTe Hepa3BepHYTble yr JIbl, o6pa30BaHHble IIPM IIepece-
qeHMM ,ZJ;BYX IIpHMbIX, eCJIM CYMMa ,ZJ;BYX M3 HMX paBHa 110° .
2. CYMMa Tpex yrJIOB, o6pa30BaHHblx IIPM IIepeCeqeHMM ,ZJ;BYX
IIpHMblX paBHa 210°. HaH,ZJ;MTe HaMMeHbIIlMH yrOJI.
3. II pM IIepeCeqeHMM ,ZJ;BYX IIpHMblX 06pa30BaJIMCb qeTblpe yr-
JIa MeHbIIle pa3BepHYTOrO. HaH,ZJ;MTe 60JIbIlIMH yrOJI, 3HaH,
qTO rpa,ZJ;YCHble Mepbl ,ZJ;BYX M3 HMX OTHOCHTCH KaK 1 : 3.
4. LCOD M LDOE - CMemHble, OM - 6McceKTpMca LCOD,
JIyq ON IIpMHa,ZJ;JIemMT BHYTpeHHeH 06JIaCTM LDOE M IIep-
IIeH,ZJ;MKY JIHpeH OM. .HBJIHeTCH JIll ON 6McceKTpMcoH LDO E?
5. BepHo JIM YTBepm,ZJ;eHMe: CYMMa BepTMKaJIbHblX yrJIOB paB-
Ha 180°?
6. M3 TOqKM 0 IIpOBe,ZJ;eHbI JIyqM OA, OB M OC, JIyq OB BHYTPII
LAOC, OB M OA - IIepIIeH,ZJ;MKYJIHpHbl. YroJI, o6pa30BaHHblH
6McceKTpMcaMM LAOBM LBOC, paBeH 60°. HaH,ZJ;MTe LBOC.
7. II pHMOH yrOJI pa3,ZJ;eJIeH JIyqOM, MCXO,ZJ;H:rn;MM M3 ero BepIIlM-
Hbl, Ha ,ZJ;Ba yrJIa, TaKMe, qTO qeTBepTb O,ZJ;Horo yrJIa paBHa
IIHTOH qaCTM ,ZJ;pyroro. HaH,ZJ;MTe 60JIbIIlMH yrOJI.
Bapuanm 4
1. HaH,ZJ;MTe Hepa3BepHYTe yr JIbl, o6pa30BaHHble IIPM rrepece-
qeHMM ,ZJ;BYX IIpHMbIX, eCJIM CYMMa ,ZJ;BYX M3 HMX paBHa 270".
2. CYMMa Tpex yrJIOB, o6pa30BaHHblx IIPM IIepeCeqeHMM ,ZJ;BYX
IIpHMblX paBHa' 250°. HaH,ZJ;MTe HaM60JIbIIlMH yrOJI.
:I. II pM IIepeCeqeHMM ,ZJ;BYX IIpHMblX 06pa30BaJIMCb qeTblpe yr-
JIB MeHbIIle pa3BepHYToro. HaH,ZJ;MTe MeHbIIlMH yrOJI, 3HaH,
'ITO rpa,ZJ;YCHble Mepbl ,ZJ;BYX M3 HMX OTHOCHTCH KaK 2 : 7.
4. / COD M LDOE - CMemHble, ON - 6McceKTpMca LDOE, JIyq
OP IIpMHa,ZJ;JIemMT BHYTpeHHeH 06JIaCTM LCOD M IIepIIeH,ZJ;M-
H:YJI.HpeH ON. .HBJIHeTCH JIM OP 6McceKTpMCOH LCOD?
ri. IJepHo JIM YTBepm,ZJ;eHMe: BepTMKaJIbHble yr JIbl paBHbl?
f5. Ha 1'OqKM 0 IIpOBe,ZJ;eHbl JIyqM OA, OB M OC, JIyq OB BHYTpl1
I AOC, OB M OA - IIepIIeH,ZJ;MKYJIHpHbl. YroJI, o6pa30BaHHbIH
fiueeeKTplIcaMII L_AOB M LBOC, paBeH 50°. HaH,ZJ;MTe L BOC.
7. PUaH( pHYTbIH yrOJI pa3,ZJ;eJIeH JIyqOM, MCXO,ZJ;HIIJ;IIM 113 ero
nnplllHHhl, H8 ABa yrJIa, 1'aKlIe, qTO qeTBepTb O,ZJ;Horo YI'JIH
PUUIIU II.H'I'OH '(neTH pyroro. H8H,ZJ;IITe MeHbIlIl1" YI'O.11.
26
CaMOe'rO.HTeJIbHaH pa60Ta NQ 12
Bapualtln I
1. TIp}lMHH /l lIepeceKaeT CTOpOHbI oCTporo LA B TOqKaX B M C.
Mory'l' JII1 ofip IIp.HMble AB M AC 6bITb rreprreH,ZJ;MKYJI.HpHbIMM
K Ill-HI MOW n.'
2. IJpH IIPI)( cplleHHM ,ZJ;BYX rrp.HMbIX o6pa30BaJIOCb qeTbIpe yrJIa
MPllhlliP pHaB( pHYToro. HaM,ZJ;MTe MeHbllIMii yrOJI, 3Ha.H, qTO
I'pH) YC 111.1(' MCpbI ,ZJ;BYX M3 HMX OTHOC.HTCH KaK 2 : 3.
:J. II I'll IH'pPCP'leHMM ,ZJ;BYX rrp.HMbIX o6pa30BaJIOCb "tJeTbIpe yr JIa.
(») UII 11 1 'IP'I'bIpex yrJIOB - rrpHMoii. MorYT JIM rpa,ZJ;YCHble Me-
phi Oe'l'UJIhHbIX Tpex yrJIOB OTHOCMTCH KaK 3 : 4 : 3?
4. / A()R 11 / BOC - CMemHbIe, LAOB Ha 20° MeHbllIe, qeM
/ R()( . JIyq OM rreprreH,ZJ;MKYJIHpeH rrpHMoii AC. Haii,ZJ;MTe
rpa) YCHYIO Mepy LBOM.
5. IlaaoBH'I'e HaH60JIbllIee qMCJIO rrpHMbIX yrJIOB, KOTOpoe MomeT
ofipaaoBaTbcH rrpl1 rrepeceqeHMM rrHTM rrpHMbIX B o,ZJ;Hoii TOqKe?
6. l epHo JIM YTBepm,ZJ;eHl1e: ,ZJ;Be rrpHMble, rreprreH,ZJ;MKYJIHpHble K
Tpe'l'heM, rrepeceKalOTcH?
7. l)aaH( pHYTbIii yrOJI pa3,ZJ;eJIeH JIyqOM, MCXO,ZJ;Hm;MM M3 ero Bep-
IIII1HhI, Ha ,ZJ;Ba yrJIa, TaKMe, "tJTO TpeTb O,ZJ;Horo yrJIa paBHa
III eC'I'OM qaCTI1 ,ZJ;pyroro. Haii,ZJ;MTe TpeTb 60JIbIIlerO yr JIa.
BalJuaum 2
1. IIpHMHH m rrepeceKaeT CTOpOHbI OCTporo LO B TOqKaX A M B.
MorYT JIM o6e rrpHMble OA M OB 6bITb rreprreH,ZJ;MKYJIHpHbIMM
K np.HMoii m?
2. llpH nepeCeqeHI1H ,ZJ;BYX rrpHMbIX o6pa30BaJIOCb qeTbIpe yrJIa
MeHbllIe pa3BepHYToro. Haii,ZJ;MTe 60JIbllIMH yrOJI, 3HaH, QTO
rpa,u,YCHble MepbI ,ZJ;BYX M3 HMX OTHOCHTCH KaK 2 : 3.
3. TIpH nepeCeqeHl111 ,ZJ;BYX rrpHMbIX o6pa30BaJIOCb qeTblpe yrJIa.
O,u,HH H3 qeTblpex yrJIOB - rrpHMoH. MorYT JIM rpa,ZJ;YCHble Me-
pbI Oe'l'aJIbHbIX Tpex yrJIOB OTHOCMTCH KaK 1 : 3 : I?
4. LAOB H LBOC - CMemHble, LAOB Ha 30° 60JIbllIe, qeM
LBOC. JIyq ON rreprreH,ZJ;I1KYJIHpeH rrpHMoii AC. Haii,ZJ;I1Te
rpa,ZJ;YCHYIO Mepy LBON.
5. Ha30BHTe HaM60JIbIIlee qHCJIO rrpHMbIX yrJIOB, KOTOpoe MomeT
06pa30BaTbCH rrpl1 rrepeceQemrn qeTb X npHMbIX B o oii TOq}{e?
6. BepHo JIll YTBepm,ZJ;eHl1e: ,ZJ;Be npHMble, rreprreH,ZJ;MKYJIHpHble K
TpeTbeM, He nepeceKalOTcH?
7. IIpHMoH yrOJI pa3,ZJ;eJIeH JIYQOM, I1CXO,ZJ;HIIJ;I1M 113 ero BepIIlI1-
HbI, Ha ,ZJ;Ba yrJIa, TaKl1e, qTO TpeTb O,ZJ;Horo yrJIa paBHa IIlec-
Toii QaCTI1 ,ZJ;pyroro. Haii,ZJ;I1Te ,lJ;Be TpeTH MeHbIIlero yr JIa.
CaMOCTOHTeJIbHaH pa60Ta N2 12
27
Bapuanm 3
1. IIpHMaH h nepeceKaeT CTOpOHbI oCTporo LA B TOqKaX B M C.
MOrYT JIM 06e npHMbIe AB M AC 6bITb nepneH,ZI;MKYJIHpHbI-
MM K npHMOH h?
2. IIpM nepeCeqeHMM ,ZI;BYX npHMbIX 06pa30BaJIOCb qeTbIpe yr-
JIa MeHbllIe pa3BepHYToro. HaH,ZI;HTe MeHbllIMH yrOJI, 3HaH,
qTO rpa,ZI;YCHbIe MepbI ,ZI;BYX H3 HHX OTHOCHTCH KaK 2 : 4.
3. IIpM nepeCeqeHMM ,ZI;BYX npHMbIX 06pa30BaJIOCb qeTbIpe yr-
JIa. O,ZI;MH M3 qeTbIpex yr JIOB - npHMoH. MorYT JIH rpa,ZI;Yc-
HbIe MepbI OCTaJIbHbIX Tpex yrJIOB OTHOCMTCH KaK 4 : 3 : 4?
4. LAOB M LBOC - CMeJKHbIe, LAOB Ha 40° MeHbllIe, qeM
LBOC. JIyq OM nepneH,ZI;MKYJIHpeH npHMoH AC. HaH,ZI;MTe
rpa,ZI;YCHYIO Mepy LBOM.
5. Ha30BMTe HaM60JIbllIee qMCJIO npHMbIX yrJIOB, KOTopoe MOJKeT
06pa30BaTbcH npH nepeCeqeHHH ceMM npHMbIX B O,ZI;HOH TOqKe?
6. BepHo JIM YTBepm,ZI;eHHe: nepneH,ZI;HKYJIHpHbIe npHMbIe -
3TO ,ZI;Be HenepeCeKalOIIJ;HeCH npHMbIe?
7. Pa3BepHYTbIH yrOJI pa3,ZI;eJIeH JIyqOM, MCXO,ZI;HIIJ;MM H3 ero Bep-
llIMHbI, Ha ,ZI;Ba yrJIa, TaKMe, qTO qeTBepTb O,ZI;HOrO yrJIa paBHa
nOJIOBMHe ,ZI;pyroro. HaH,ZI;MTe qeTBepTb MeHbllIero yrJIa.
Bapuanm 4
1. IIpHMaH b nepeceKaeT CTOpOHbI OCTporo LO B TOqKaX A M B.
MorYT JIH 06e npHMbIe OA H OB 6bITb nepneH,ZI;MKYJIHpHbI-
MM K npHMoH b?
2. IIpM nepeCeqeHMM ,ZI;BYX npHMbIX 06pa30BaJIOCb qeTbIpe yr-
JIa MeHbllIe pa3'BepHYToro. HaH,ZI;MTe 60JIbllIHH yrOJI, 3HaH,
qTO rpa,ZI;YCHbIe MepbI ,ZI;BYX M3 HMX OTHOCHTCH KaK 1 : 5.
3. IIpM nepeCeqeHMM ,ZI;BYX npHMbIX 06pa30BaJIOCb qeTbIpe yr-
JIa. O,ZI;MH M3 qeTbIpex yr JIOB - npHMoH. MorYT JIM rpa,ZI;Yc-
HbIe MepbI OCTaJIbHbIX Tpex yrJIOB OTHOCMTCH KaK 2 : 3 : 2?
4. LAOB M LBOC - CMemHbIe, LAOB Ha 50° 60JIbllIe, qeM
LBOC. JIyq ON nepneH,ZI;MKYJIHpeH npHMoH AC. HaH,ZI;MTe
rpa,ZI;YCHYIO Mepy LBON.
5. Ha30BMTe HaM60JIbllIee qMCJIO npHMbIX yrJIOB, KOTopoe MomeT
06paaoBaTbcH npH nepeceqeHMMllIecrn npHMbIX B O,ZI;HOH TOqKe?
6. BepHo JIM YTBepm,ZI;eHMe: nepneH,ZI;MKY JIHpHbIe npHMbIe -
3TO ,ZI;Be nepeCeKalOIIJ;MeCH npHMbIe, KOTopbIe 06pa3YlOT qe-
TbIpe npHMbIx yr JIa ?
7. IIpHMoH yrOJI pa3,ZI;eJIeH JIytIOM, MCXO,ZI;HIIJ;MM M3 ero BepllIM-
HbI, Ha ,ZI;Ba yrJIa, TaKMe, qTO qeTBepTb O,ZI;HOrO yrJIa paBHa no-
JIOBMHe ,ZI;pyroro. HaH,ZI;MTe TPH qeTBepTM 60JIbllIerO yr JIa.
28
CaMOCTOHTeJIbHaH pa60Ta N2 13
Bapuanm 1
1. If3 BepllIMHbI pa3BepHyroro yr JIa npOBe,n;eHbI ,n;Ba .rryqa, KOTopbIe
,n;eJIHT ero Ha TPM paBHbIe QaCTI1. IIepneH,lJ;I1KYJIHpHa JII16I1CCeK-
Tplica cpe,n;Hero yr JIa CTopoHaM pa3BepHYTOrO yr JIa ?
2. Haii,n;I1Te yrOJI, o6pa30BaHHbIii 611CCeKTpl1CaMI1 ,lJ;BYX CMem-
HbIX yrJIOB.
3. JIemaT JIM 6McceKTpHCbI BepTI1KaJIbHbIX yrJIOB Ha o,n;Hoii
rrpHMon?
4. 113 TOqKM 0 IlpOBe,lJ,eHbI JIyqM OA, OB 11 OC, JIyq OB rrepneH-
,n;MKYJInpeH JIyt.IY OA, a Jlyq OC - BHYTpl1 LAOB. YroJI, o6pa-
30BaHHbIH 6Hceel\:TpHcaMH AOB 11 LBOC, paBeH 20".
Han HTe / BOC.
5. II pnMoM yrOJI paal eJIeH JIY'-IOM, HCXO,lJ;nm;IIM 113 ero BepllIM-
HbI, Ha ,n;Ba yrJIa, TaKHe, '-ITO rrOJIOBMHa o,n;Horo yrJIa paBHa
TpeTM ,n;pyroro. Ha CKOJIbKO rpa,n;YCOB O,n;MH 113 3TI1X yr JIOB
MeHbllie ,n;pyroro?
6. Ha CKOJIbKO qacTeii pa3611BalOT nJIOCKOCTb nHTb rrpHMbIX,
nepeCeKalOIIJ;I1XCH B o,n;Hoii TOqKe?
7. CYMMa ,n;BYX yr JIOB TpeyrOJIbHI1Ka paBHa 130°, a O,n;I1H 113 3TI1X
yr JIOB 60JIbllIe ,n;pyroro Ha 70°. Haii,n;I1Te 60JIbllII1ii yrOJI Tpe-
yrOJIbHI1Ka.
Bapuanm 2
1. 113 BepllII1HbI pa3BepHYTOrO yr JIa rrpOBe,n;eHbI qeTbIpe .rryqa, KO-
TopbIe ,n;eJIHT ero Ha rrHTb paBHbIX qacTeii. IIeprreH,n;I1KY JIHpHa
JII1 611cceKTpl1ca cpe,n;Hero yr JIa CTopoHaM pa3BepHYTOrO yr JIa?
2. Haii,n;I1Te yrOJI, o6pa30BaHHbIii 611CCeKTpl1CaMI1 ,n;BYX BepTI1-
KaJIbHbIX yrJIOB.
3. JIemaT JII1 6lICCeKTpl1CbI CMemHblX yr JIOB Ha rreprreH,n;I1KY-
JIHpHbIX rrpHMbIx?
4. 113 TOqKI1 0 rrpOBe,n;eHbI JIyql1 OC, OD 11 OE, JIyq OC rreprreH-
,n;I1KYJIHpeH JIyqy OD, a JIyq OE - BHYTpl1 LCOD. YroJI,
o6pa30BaHHbIii 611CCeKTpl1CaMI1 LCOD 11 LEOD, paBeH 25°.
Haii,n;I1Te LEOD.
5. Pa3BepHYTbIii yrOJI pa3,n;eJIeH JIyqOM, I1CXO,n;HIIJ;I1M 113 ero Bep-
IlII1HbI, Ha ,lJ;Ba yrJIa, TaKMe, qTO rrOJIOBI1Ha o,n;Horo yrJIa paBHa
TpeTII ,n;pyroro. Ha CKOJIbKO rpa,n;YCOB O,n;I1H 113 3TMX yr JIOB
60JIbllIe ,n;pyroro?
6. Ha CKOJIbKO qacTeii pa3611BalOT rrJIOCKOCTb llIeCTb rrpHMbIX,
rrepeCeKalOIIJ;IIXCH B o,n;Hoii TOqKe?
7. CYMMa ,n;BYX yr JIOB TpeyrOJIbHI1Ka paBHa 120°, a O,n;I1H 113 3TI1X
yr JIOB MeHbllIe ,n;pyroro Ha 50°. Han,n;I1Te MeHbllIl1ii yrOJI Tpe-
yrOJIbHI1Ka.
CaMOCTOHTeJIbHaH pa60Ta NQ 13
29
Bapuaum 3
1. 11:3 :aepI.illlHbI paasepHY'fOro yrJIa rrpoOO,n;eHbI llIecTb .rryqeH, KaropbIe
,n;eJIHT ero Ha ceMb paBHbIX qaCTeH. IIeprreH,n;IIKY JIHpHa JIM
6IICCeKTpMca cpe,n;Hero yrJIa CTopOHaM pa3BepHYToro yrJIa?
2. KaKOBa rpa,n;YCHaH Mepa yrJIa, o6pa30BaHHoro 6McceKTpM-
caMM ,n;BYX CMemHbIX yr JIOB?
3. BepHo JIM YTBepm,n;eHMe: 6McceKTplICbI BepTIIKaJIbHbIX yrJIOB
JIemaT Ha O,n;HOM rrpHMoM?
4. 11:3 TOqKM 0 rrpOBe,n;eHbI JIyqM OA, OB II OC, JIyq OB rreprreH-
,n;MKYJIHpeH JIyqy OA, a JIyq OC - BHYTPM LAOB. YroJI, o6pa-
30BaHHbIM 6McceKTpMcaMM LAOB M LBOC, paBeH 30°.
HaM,n;MTe LBOC.
5. IIpHMoM yrOJI pa3,n;eJIeH JIyqOM, MCXO,n;Hm;MM M3 ero BepIIIM-
HbI, Ha ,n;Ba yrJIa, TaKMe, qTO rrOJIOBIIHa o,n;Horo yrJIa paBHa
qeTBepTII ,n;pyroro. Ha CKOJIbKO rpa,n;YCOB O,n;MH M3 3TMX yrJIOB
MeHbIIIe ,n;pyroro?
6. Ha CKOJIbKO qaCTeM pa36MBalOT rrJIOCKOCTb TpII rrpHMbIe,
rrepeceKalOm;MXCH B O,n;HOM TOqKe?
7. CYMMa ,n;BYX yr JIOB TpeyrOJIbHMKa paBHa 110°, a O,n;MH M3 3TMX
yr JIOB 60JIbIIIe ,n;pyroro Ha 60°. HaM,n;IITe 60JIbIIIIIM yrOJI Tpe-
yrOJIbHMKa.
Bapuaum 4
1. 11:3 OOpI.illlHbI paaOOPHY'fOro yrJIa rrpooo,n;ellliI BOCeMb .rryqeM, KaropbIe
,n;eJIHT ero Ha ,n;eBHTb paBHbIX qaCTeM. IIeprreH,n;MKY JIHpHa JIM
6IICCeKTpMca cpe,n;Hero yrJIa CTopOHaM pa3BepHYToro yrJIa?
2. KaKOBa rpa,n;YCHaH Mepa yrJIa, o6pa30BaHHoro 6McceKTpM-
caMM ,n;BYX BepTMKaJIbHbIX yrJIOB?
3. BepHo JIM YTBepm,n;eHMe: 6McceKTpMCbI CMemHbIX yrJIOB
JIemaT Ha rreprreH,n;MKYJIHpHbIX rrpHMbIx?
4. 11:3 TOqKII 0 rrpOBe,n;eHbI JIyqlI OC, OD II OE, JIyq OC rreprreH-
,n;IIKYJIHpeH JIyqy OD, a JIyq OE - BHYTPM LCOD. YroJI,
o6pa30BaHHbIM 6McceKTpMcaMM LCOD M LEOD, paBeH 35°.
HaM,n;MTe LEOD.
5. Pa3BepHYTbIM yrOJI pa3,n;eJIeH JIyqOM, MCXO,n;Hm;IIM M3 ero Bep-
1lIIIHbI, Ha ,n;Ba yrJIa, TaKlIe, qTO rrOJIOBIIHa o,n;Horo yrJIa paBHa
qeTBepTM ,n;pyroro. Ha CKOJIbKO rpa,n;YCOB O,n;IIH 113 3TIIX yr JIOB
60JIbIIIe ,n;pyroro?
6. Ha CKOJIbKO qaCTeM pa36MBalOT rrJIOCKOCTb qeTbIpe rrpHMbIe,
rrepeceKalOm;MXCH B O,n;HOM TOqKe?
7. CYMMa ,n;BYX yr JIOB TpeyrOJIbHMKa paBHa 100°, a O,n;IIH M3 3TIIX
yrJIOB MeHbIIIe ,n;pyroro Ha 40°. Haii,n;MTe MeHbIIIMM yrOJI Tpe-
yrOJIbHMKa.
30
CaMoeTO I'reJIbHaH pa60Ta N2 14
Bapuanm I
1. ABI) ;\( HJ), HD - MX 06m;aH CTopOHa, AD = DC, LABC =
--= 11 () , / RA D -= 90°. HaH MTe LBDC.
2. B ;\Al ( AR BC, AC = 8 CM, TO Ka E rrpMHa JIemMT BC,
I E f;( . 'rotlKM A H E eJIHT rrepMMeTp ABC Ha Be qaCTM,
H:J I<:OTOphIX O Ha 60JIbIIIe pyroH Ha 2 CM. HaH MTe AB.
:J. B ;\A I C AB = BC = CA. BHYTPM TpeyrOJIhHMKa BhI6paHa
T()\Il<:U () TaK, qTO LAOB = LAOC = 120". HBJIHeTCH JIM AO
fiHcceKTpMcoH LBAC?
4. 13epHo JIM YTBepm eHlie: rrepMMeTp TpeyrOJIbHliKa - rrpOM3-
HeAeHMe JIMH Tpex CTOpOH TpeyrOJIhHMKa?
5. IIepMMeTp O Horo TpeyrOJIhHMKa Ha 1 CM 60JIhIIIe rrepMMeTpa
,n;pyroro TpeyrOJIhHMKa. MorYT JIM 6bITh paBHbIMM 3TH Tpe-
yrOJIbHMKH?
6. IIepMMeTp TpeyrOJIhHMKa paBeH 9 CM. O Ha CTopOHa B 2 pa3a
60JIbIIIe pyroH M Ha 1 CM 60JIhIIIe TpeTheH. HaM MTe
MeHbIIIYIO CTOPOHY TpeyrOJIbHMKa.
7. AaH C, TOqKa D rrpMHa,ZJ;JIemMT BC, a TOqKa E - AC. OT-
pe3KM AD M BE rrepeceKalOTCH B TOqKe O. CKOJIbKO Bcero Tpe-
yrOJIbHHKOB?
Bapuanm 2
1. BC = BCD, BC - MX 06m;aH CTopOHa, AB = BD, LACD =
= 130°, LBAC = 90°. HaH MTe LCBD.
2. B BC AB = BC, AC = 7 CM, TOqKa N rrpMHa JIemMT BC,
BN = NC. TOqKM A M N eJIHT rrepMMeTp C Ha Be qaCTM,
M3 KOTOphIX O Ha 60JIhIIIe ,n;pyroH Ha 1 CM. HaH HTe BC.
3. B ABC AB = BC. BHYTPM TpeyrOJIbHMKa BbI6paHa TOqKa K
1'aK, TO LAKB = LAKC = 130°. HBJIHeTCH JIM AK 6McceK-
'rpMCOH LBAC?
4. BepHo JIH YTBepm eHMe: paBHbIX TpeyrOJIhHMKaX rrpOTMB
COOTBeTCTBeHHO paBHhIX CTOpOH JIemaT paBHbIe yrJIhI?
5. llepMMeTp O Horo TpeyrOJIbHMKa Ha 2 CM MeHbIIIe rrepMMeT-
pa ,n;pyroro TpeyrOJIhHMKa. MorYT JIM 6hITh paBHhIMM 3TM
TpeyrOJIbHMKM?
6. IIepMMeTp TpeyrOJIbHMKa paBeH 13 CM. O Ha CTopOHa B 3 pa3a
60JIhIIIe pyroH Ii Ha 1 CM 60JIbIIIe TpeTbeH. HaH MTe MeHb-
IIIYIO CTOPOHY TpeyrOJIhHMKa.
7. ,1J;aH ABC, TOtIKa N rrpMHa JIemMT AB, a TOqKa M - AC. OT-
pe3KH CN M BM rrepeceKalOTCH B TOqKe o. CKOJIhKO Bcero
TpeyrOJIhHMKOB?
CaMOCTOHTeJlbHaH pa60Ta N2 14
31
Bapuanm 3
1. LlABD = /).CBD, BD -I1X o6 aH CTopOHa, AD = DC, LABC =
= 100°, LBAD = 90°. HaH,l\I1Te LBDC.
2. B LlABC AB = BC, AC = 6 CM, TO'-lKa E rrpl1Ha,l\JIeml1T BC,
BE = EC. TO'IKI1 A 11 E ,l\eJIHT nepHMeTp /).ABC Ha ,l\Be 'IaCTI1,
113 KOTOpbIX O,l\Ha 60JIbIlle ,l\pyroii Ha 2 CM. HaH,l\I1Te AB.
3. B /).ABC AB = BC = CA. BHYTPH TpeyrOJIbHHKa BbI6paHa
TO'IKa D TaK, 'ITO L_ADB = LADC = 120'. HBJIHeTCH JIM AD
611cceKTpMcoH LBAC?
4. BepHo JIM YTBepm,l\eHl1e: nepl1MeTp TpeyrOJIbHMKa - CYMMa
,l\JIMH Tpex CTOpOH TpeyrOJIhHHKa?
5. IIepl1MeTp O,l\HOrO TpeyrOJIbHMKa Ha 2 CM 60JIbIIIe nepl1MeTpa
,l\pyroro TpeyrOJIbHI1Ka. MorYT JIM 6bITb paBHbIMI1 3TI1 Tpe-
yrOJIbHI1KI1?
6. IIepl1MeTp TpeyrOJIbHI1Ka paBeH 13 CM. O,l\Ha CTopOHa B 2 pa-
3a 60JIbIIIe ,l\pyroH 11 Ha 2 CM 60JIbIIIe TpeTbeH. Haii,l\I1Te
MeHbIIIYIO CTOPOHY TpeyrOJIbHI1Ka.
7. ,l];aH BC, TO'IKa N npl1Ha,l\JIemMT BC, a TO'IKa M - AC. OT-
pe3KM AN 11 BM nepeceKaIOTCH B TO'-lKe o. CKOJIbKO Bcero
TpeyrOJIbHMKOB?
Bapuanm 4
1. LlABC = /).BCD, BC - I1X o6 aH CTopOHa, AB = BD, LACD =
= 120°, LBAC = 90°. Haii,l\I1Te LCBD.
2. B /).ABC AB = BC, AC = 9 CM, TO'-lKa N npl1Ha,l\JIeml1T BC,
BN = NC. TO'IK A M N ,l\eJIHT nepl1MeTp C Ha ,l\Be 'IaCTI1,
113 KOTOpbIX O,l\Ha 60JIbIIIe ,l\pyroii Ha 3 CM. HaH,l\I1Te BC.
3. B C AB = BC. BHYTPM TpeyrOJIbHHKa BbI6paHa TO'IKa M
TaK, 'ITO LAMB = LAMC = 140°. HBJIHeTCH JII1 AM 6l1cceK-
Tpl1COH LBAC?
4. BepHo JII1 YTBepm,l\eHl1e: paBHbIe TpeyrOJIbHI1KI1 MomHO
COBMeCTI1Tb HaJIOmeHl1eM?
5. IIepl1MeTp O,l\HOro TpeyrOJIbHI1Ka Ha 1 CM MeHbIIIe nepl1MeT-
pa ,l\pyroro TpeyrOJIbHI1Ka. MorYT JII1 6bITb paBHbIMH 3TI1
TpeyrOJIbHI1KI1?
6. IIepMMeTp TpeyrOJIbHI1Ka paBeH 18 CM. O,l\Ha CTopOHa B 3 pa-
3a 60JIbIIIe ,l\pyroH 11 Ha 3 CM 60JIbIIIe TpeTbeH. HaH,l\I1Te
MeHbIIIYIO CTOPOHY TpeyrOJIbHHKa.
7. ,l];aH /).ABC, TO'IKa E npl1Ha,l\JIemMT AB, a TO'-lKa F - AC. OT-
pe3KI1 CE 11 BF nepeceKaIOTCH B TO'IKe o. CKOJIbKO Bcero Tpe-
yrOJIbHI1KOB?
32
CaMOCTOHTeJIbHaH pa60T3 N2 15
Bapuanm 1
1. OTpe3KM AB M CD TO"tIKOH rrepece-qeHH.H 0 tJ;eJIJITC.H nOIIO-
.rraM. BY,l\YT JIM paBHbI AOC Ii !1BOD?
2. IIepMMeTp O,l\HOrO TpeyroJlbHMKa MeHbIIIe Ha 2 CM IIepMMeT-
pa tJ;pyroro. MOrYT JIM 6bITh paBHbIMM 31'}1 TpeyrOJIbHMKM'!
3. ,[(aH /\ABC. T01:IKR D ME JIema'f Ha cTOpoHe AC. LBDC pa-
BeH LBEA, AD = EC, BD = BE, LBCE paBeH 40 . qeMY pa-
BeH LBAD?
4. )];aH AABC, TOllKa D JIemlIT BHe AABC, AB = BC = BD = DC.
IIepMMeTp L\ABC paBeH 42 CM, AC = 12 CM. HaH HTe nepM-
MeTp t1BDC.
5. aH t1ABC, TOqRa K JIemI1T Ha CTopOHe AC, TOqRa 0 - Ha
OTpe3Ke BK, OA = OC, LAOB paBeH LBOC. BY,l\eT JIM
AOK paBeH COK?
6. aH !1ABC, AB = BC = 12 CM. TOQK3 M JIemMT Ha CTopOHe
BC, npMqeM BM = MC. HaH,n;llTe CTOPOHY AC, eCJIH OHa B
2 pa3a MeHbllIe, qeM BM.
7. B !1ABC BhI6paHa T04K3 0 TaK, QTO AOB paBeH COB,
OA = OC, LAOC = 140 . HaHtJ;JITe LAOB.
Bapuanm 2
1. OTpe3KM AB M CD TOqROH nepeCeqeHHH 0 ,n;eJlHTCH IIorro-
JIaM. By,n;YT JIM paBHbI AOD If !1BOC?
2. IIepMMeTp O}J;Horo TpeyrOJIbHMRa 60JIbllIe Ha 3 CM nepHMeT-
pa ,n;pyroro. MorYT JIM 6hITb paBHhIMM 9TM TpeyroJlbHIIKH?
3. J-:(aH !\ABC. TOqKM D H E JIemaT Ha CTopOHe AC. LBDC pa-
BeH LBEA, AD = EC, BD = BE, LBAD paBeH 30 . qeMY pa-
BeH LBCE?
4. ,L{aH ACDE, TOQKa F JIeJRMT BHe !\CDE, CD = DE = DF = EF.
IIepMMeTp CDE paBeH 37 CM" CE = 13 CM. HaH HTe rrepn-
MeTp DEF.
5. aH L\ABC, TOqKa M .nemMT Ha CTopOHe AC, TO"[IKa 0 - Ha
OTpe3Ke BIVI, OA = OC, L_AOB paBeH LBOC. By eT .HH
AO}\l paBeH ACOM?
6. aH ABC, AB -== BC -= 18 CM. ToqK3 K JlemHT Ha CTopOHe
BC, IIpnlJeM BK = KC. HaHt\lfTe CTOPOHY AC, eCJIM OHa B 3
pa3a MeHblue, qeM BK.
7. B ABC BhI6paHa TOqRa 0 TaR, QTO t1AOB paBeH ACOB,
OA = OC, / AOC = 160 . IIai1 MTe BOC.
CaMOCTO.f.lTeJlbH3H pa60Ta N 15
33
Bapuanm 3
1. OTpe3Klf CD M EF TOtJKOH nepeCeqeHM.H 0 ,n;eJI.f.ITC.f.I rrorro-
JlaM. By,n;YT JIll paBHhI I1COF H I1DOE?
2. ITepllMeTp O)];HOrO TpeyrOJIbHMKa MeHbllle H3 5 CM rrepHMeT-
pa ,n;pyroro. MorYT JIll 6bITb paBHhIMll 3TH TpeyrOJIbHMKll?
3. ,lJ:aH L\ABC. TOqKII D II E JIemaT Ha CTopOHe AC. LBDC pa-
BeH LBEA, AD = EC, BD = BE, LBCE paBeH 50:>. LIeMY pa-
BeH LBAD?
4. ,1];an iVU3C, T01:IKa D JIemHT BHe I1ABC, AB = BC = BD = DC.
ITepllMeTp !\ABC paBeH 39 CM, AC = 1 7 CM. HaH,lJ;HTe nepll-
MeTp I1BDC.
5. ,1];aH I1ABC, To"tJKa N JIemllT Ha CTopOHe BC, To"tJKa 0 - Ha
OTpe3Ke AN, OB = OC, LAOB paBeH LAOC. By,n;eT JIll
I1CON paBeH I1BON?
6. ,1];aH MBC, AB = BC = 16 CM. TO"tJKa D JIemHT Ha CTopoHe
BC, rrpHqeM BD = DC. Hall,lJ;IITe CTOPOHY AC, eCJIH oHa B
2 pa3a MeHbllIe, qeM BD.
7. B I1ABC BbI6paHa TOqKa 0 TaK_ qTO AOB paBeH !\COB,
OA = OC, LAOC = 130'. Han)];I1Te / AOB.
Bapuanm 4
1.0Tpe3KII CD II EF TOqKOH rrepeCeqeHHH 0 ,n;eJIHTCH Ilono-
JIaM. By,n;YT JIM paBHbI COE If !1DOF?
2. ITepMMeTp O)];HOrO TpeyrOJIbHHKa 6o.Hbllle Ha 4 CM nepMMeT-
pa ,n;pyroro. MorYT JIM 6bITb paBHbIMM 3TH TpeyroJlbHMKM?
3. ,1];aH I1ABC. TOtJKM D M E JIemaT Ha CTopOHe AC. LBDC pa-
BeH LBEA, AD = EC, BD ==- BE, LBAD paBeH 20 . qeMY pa-
BeH LBCE?
4. ,1];aH CDE, TotIKa F Jleml1T BHe I1CDE, CD = DE = DF = EF.
ITepllMeTp I1CDE paBeH 36 CM, CE =- 10 CM. HaH,l);IITe rrepIf-
MeTp DEF.
5. ,1];aH ABC, To"tJKa K Jle:iKMT Ha CTopOHe BC, T01.J:Ka 0 -- Ha
OTpe3Ke AK, DB = OC, LAOB paBeH L_AOC. BYj(eT J1I1
I1COK paBeH BOK?
6. ,1];aH I1ABC, AB = BC = 24 CM. ToqK3 N JlCml1T Ha eTopoHe
BC, rrpHqeM BN = NC. HaH,n;MTe CTOPOHY AC, eCJ111 OHa B
3 pa3a MeHbIIIe, qeM BN.
7. B I1ABC BhI6paHa TOqKa 0 TaK, TO L\AOB paB8H L\COB,
OA = OC, LA()C = 150 . HaH,lJ;MTe /_BOC.
34
CaMOCTOHTeJIbHaH pa60Ta N 16
Bapuanm J
1. OTpe3Kl1 AB 11 CD rrepeceKalOTCH B TOqKe 0, HBJIHIOrn;enCH
cepe,lJ;MHOM }{8m,lJ;OrO 113 HMX. Han,lJ;MTe LOAC, eCJIM LOBD
Ha 10 60JIhllle, qeM LBOD 11 Ha 40° MeHbllIe, qeM LODB.
2. TOqKM A M B JIemaT no O,lJ;HY CTOPOHY OT npHMon a. I1epneH-
,lJ;l1KYJI}{PhI AC M BD K npHMon a paBHbI. Hall,lJ;MTe LCAD,
eCJIH / BCD = 42°.
3. Ha CTopOHax LAOB OTMeqeHbI TOqKM C H D TaK, qTO TOqKa
( JI( mHT Ha OTpe3Ke OA, a TOqKa D - Ha OTpe3Ke OB, OA =
=- OB M OC = OD. CpaBHMTe LACB M LBDA.
4. II Plep'rHTe npHMYIO a. OTMeTbTe TOqKY A, He JIemarn;ylO Ha
IIP}{MOH a. CKOJIbKO nepneH,lJ;MKYJIHpOB MomHO npOBeCTM M3
TOlIKM A K npHMon a?
5. ,l],HHhI ,lJ;Be nepeceKalOrn;MeCH npHMbIe a H b M TOqKa A, He JIe-
maIlJ;8H Ha STMX npHMbIX. Qepe3 TOqKY A npOBe,lJ;eHbI npHMbIe
/1211 n TaK, qTO m-La M n-Lb. COBna,n;aIOT JIM npHMbIe m M n?
6. BepHO JIM YTBepm,lJ;eHMe: M3 TOqKM, He JIemarn;en Ha npH-
MOM, MomHO npoBecTM nepneH,lJ;MKYJIHp K sTon npHMon, M
npMToM TOJIbKO O,lJ;MH?
7. MorYT JIM CTOpOHbI TpeyrOJIbHMKa 6bITb 1 CM, 2 CM, 3 CM?
Bapuanm 2
1. OTpe3KM AB M CD nepeceKalOTCH B TOqKe 0, HBJIHIOrn;enCH
Cepe,lJ;MHOn Kam,lJ;OrO M3 HMX. Han,lJ;MTe LACO, eCJIM LOBD
Ha 20" 60JIbllIe, qeM LBOD M Ha 20° MeHbllIe, qeM LODB.
2. TOqKM C M D JIemaT no O,lJ;HY CTOPOHY OT npHMon a. I1epneH-
,lJ;MKYJIHpbI CE 11 DF K npHMon a paBHbI. Hall,lJ;MTe LEDF,
eCJIM LCFE = 46°.
3. Ha CTopOHax LAOB OTMeqeHbI TOqKM N M M TaK, qTO TOqKa
N JIemMT Ha OTpe3Ke OA, a TOqKa M - Ha OTpe3Ke OB, OA =
= OB M ON = OM. CpaBHMTe LANB M LAMB.
4. HaqepTHTe npHMYIO b. OTMeTbTe TOqKY B, He JIemarn;ylO Ha
npHMon b. CKOJIbKO nepneH,lJ;IIKYJIHpOB MomHO npOBeCTM M3
TOqKM B K npHMon b?
5. Qepe3 TOqKY N, He JIemam;ylO Ha npHMOll n, npOBe,lJ;eHbI qeTbI-
pe npHMhIe, nepeceKalOrn;Me npHMYIO n. BepHO JIM, qTO no
KpaHHen Mepe TPM M3 HMX He nepneH,lJ;MKYJIHpHbI K npHMon n?
6. BepHo JIM YTBepm,lJ;eHMe: M3 TOqKM, He JIemarn;ell Ha npHMon,
MomHO npOBeCTM ,lJ;Ba nepneH,lJ;MKYJIHpa K sTon npHMon?
7. MorYT JIM CTOpOHbI TpeyrOJIbHMKa 6bITb 2 CM, 3 CM, 3 CM?
CaMOCTOHTeJIbHaH pa60Ta N2 16
35
Bapuaum 3
1. OTpe3KM AB M CD nepeceKalOTCH B TOqKe 0, HBJIHIOm;eHCH
cepe MHoH Kam oro M3 HHX. HaH HTe LOAC, eCJIH LOBD
Ha 10° 60JIbllIe, qeM LBOD M Ha 10(> MeHbllIe, qeM LODB.
2. TOqKM A M B JIemaT no O,l];HY CTOPOHY OT npHMoH a. IIepneH-
HKYJIHpbI AC M BD K npHMoH a paBHbI. HaH,l];MTe LCAD,
eCJIM LBCD = 44".
3. Ha CTopOHax LAOB OTMeqeHbI TOqKM C M D TaR, qTO TOqKa
C JIemMT Ha OTpe3Ke OA, a TOqKa D - Ha OTpe3Ke OB, OA =
= OB M OC = ODe CpaBHMTe LCAD H LCBD.
4. HaqepTMTe npHMYIO n. OTMeTbTe TOqRY N, He JIemam;ylO Ha
npHMOH n. CKOJIbKO nepneH,l];MKYJIHpOB MomHO npOBeCTM M3
TOqKM N K npHMOM n?
5. ,lJ;aHbI Be nepeceKalOm;MeCH npHMbIe a M b M TOqKa B, He JIe-
mam;aH Ha 3TMX npHMbIX. Qepe3 TOqKY B npOBe eHbI npHMbIe
n H m TaK, qTO n-La M m-Lb. COBnMalOT JIM npHMbIe n M m?
6. BepHO JIM YTBepm eHMe: M3 TOqRM, He JIemam;eH Ha npH-
MOM, MomHO npoBecTM nepneH MKYJIHp R 3TOM npHMOM, M
npMToM TOJIbKO O HH?
7. MorYT JIM CTOpOHbI TpeyrOJIbHMKa 6bITb 1 CM, 3 CM, 4 CM?
Bapuaum 4
I. OTpe3KM AB M CD nepeceKalOTCH B TOqKe 0, HBJIHIOm;eHCH
cepe HHoH Kam oro H3 HMX. HaH MTe LACO, eCJIM LOBD
Ha 10° 60JIbllIe, .qeM LBOD M Ha 40° MeHbllIe, qeM LODB.
2. TOqKM C M D JIemaT no O HY CTOPOHY OT npHMoM a. IIepneH-
HKYJIHpbI CE M DF K npHMoH a paBHbI. HaH MTe LEDF,
eCJIH LCFE = 38°.
a. Ha CTopOHax LAOB OTMeqeHbI TOqKM N M M TaK, qTO TOqKa
N JIemMT Ha OTpe3Ke OA, a TOqKa M - Ha OTpe3Ke OB, OA =
= OB M ON = OM. CpaBHMTe LNAM M LNBM.
4. HaqepTHTe npHMYIO m. OTMeTbTe TOqKY M, He JIemam;ylO
Ha npHMOH m. CKOJIbKO nepneH MKYJIHpOB MomHO npOBec-
TH H3 TOqKM M K npHMOH m?
5. Qepe3 TOqKY A, He JIemam;ylO Ha npHMOH a, npOBe,l];eHbI TPM
npHMbIe, nepeceKalOm;Me npHMYIO a. BepHO JIM, qTO no KpaM-
HeH Mepe Be M3 HHX He nepneH MKYJIHpHbI K npHMOH a?
6. BepHO JIM YTBepm eHHe: M3 TOqKM, He JIemam;eH Ha npHMOH,
MomHO npOBeCTM Ba nepneH MKYJIHpa K 3TOM npHMoM?
7. MorYT JIM CTOpOHbI TpeyrOJIbHMKa 6bITb 1 CM, 3 CM, 3 CM?
36
CaMOCTOHTeJIbHaH pa60Ta NQ 1 7
Bapuanm 1
1. HaqepTHTe TpeyrOJIbHHK, rrpOBe,l);HTe Me,l);HaHhI. IIepeceKa-
IOTCH JIH TPM Me.n;HaHbI B O.n;HOH TO'tlKe?
2. BepHo JIH YTBepm eHHe: 6HcceKTpMca TpeyrOJIbHHKa - OTpe-
30K 6McceKTpHchI yrJIa TpeyrOJIbHHKa, COe,lJ;MHSIIOID;HH Bep-
mHHY TpeyrOJIbHHKa C TO'tlKOH rrpoTMBOnOJIOmHOH CTOpoHhI?
3. HaqepTHTe oCTpoyrOJIbHbIM TpeyrOJIbHHK, npOBe.n;HTe BbI-
COThI. TO'tlKa nepeCe"lleHHH BbICOT pacnOJIO.iKeHa BO BHYT-
peHHeH 06JIaCTH TpeyrOJIbHHKa?
4. MomeT JIH CTopOHa TpeyrOJIbHHKa COBna.n;aTb C BbICOTOM
TpeyrOJIbHHKa?
5. HaLIepTHTe TynoyrOJIhHbIH MBC, LB - TynoH. IIpoBe.n;HTe
Me.n;HaHY AN H BLICOTY AH. KaKoM H3 OTpe3KOB HaHMeHh-
illMH AH, AN HJIH AB?
6. Ha'tlepTHTe oCTpoyrOJIhHhIM ABC, AB = AC = 4 CM. IIpoBe-
,lJ;HTe Me,ZJ;HaHY AO H 6McceKTpHcy AE. HaH,lJ;HTe .n;JIHHY OT-
pe3Ka OE.
7. B ABC BbI6paHa TO'tlKa 0 TaK, LITO AOB paBeH t).COB,
OA = OC, LBOC = 100°. Haii.n;MTe LAOC.
Bapuanm 2
1. Ha'tlepTHTe TpeyrOJIbHHK, npOBe.n;MTe 6HcceKTpHcbI. IIepece-
KaIOTCH JIH TpH 6HcceKTpHcbI B O,lJ;HOH TOQKe?
2. BepHo JIB YTBepm.n;eHHe: Me)J;MaHa TpeyrOJIbHMKa - OTpe-
30K, coe,ZJ;MHHIOru;BH BepmHHY TpeyrOJIbHHKa C cepe.n;HHOH
npOTHBOnOJIO.iKHOH CTOpOHhI?
3. HaqepTHTe TynoyrOJIbHbIH TpeyrOJIbHMK, npOBe.n;HTe BbICO-
Tbl. TO'tlKa nepeCeQeHHH rrpo.n;OJIJKeHMH BbICOT Haxo.n;HTCH
BO BHeIIIHeH 06JIaCTH TpeyrOJIbHHKa?
4. MomeT JIM CTopOHa TpeyrOJIbHHKa COBIIap;aTb C 6McceKTpH-
COM TpeyrOJIhHHKa?
5. HaLIepTMTe oCTpoyrOJIbHbIM pa3HocTopOHHHH MBC. IIpo-
Be.n;HTe Me,ZJ;HaHY AO H BbICOTY AH. KaKoH H3 OTpe3KOB HaH-
60JIbillHH AH, AO MJIH AC?
6. HatIepTHTe oCTpoyrOJIbHhIH CDE, CD = DE = 5 CM. IIpo-
Be,n;HTe Me,lJ;HaHY DO H 6HcceKTpHcy DN. H ii,ZJ;HTe ,lJ;JIHHY
OTpe3Ka ON.
7. B ABC BbI6paHa TO'tlKa 0 TaR, tITO AOB paBeH COB,
OA = OC, LAOB ; 110°. HaM)J;HTe LAOC.
CaMOCTOHTeJIbHaH pa60Ta N2 1 7
37
BapU4Hm 3
1. HaqepTHTe TpeyrOJlbHHK, rrpoBeAHTe Me,lJ;HaHbI. Bcer,n;a JIB
TOQKa nepeCe'tJeHHH Me HaH HaxoAHTCH BO BHYTpeHHeH 06-
JlaCTH TpeyrOJIbHHKa?
2. BepHo JlH YTBepm,lJ;eHHe: JII060H TpeYI'OJIhHHK HMeeT TPH
6HcceKTpHcbI?
3. BepHo JIH yrBepm,lJ;eHHe: B oCTpoyroJlbHoM TpeyrOJIbHHKe
TO:qJ((\ nepeCeqeHHH BLICOT HaXO,lJ;HTCH BO BHYTpeHHeii 06-
JIaCTH TpeyrOJlLHHKa?
4. MomeT JIH BLICOTa TpeyroJlbHHKa COBn aTb co CTOpOHOH
TpeyroJIbHHKa ?
5. HaqepTHTe TynoYI'OJILHLIH MBC, LB - TynoH. IIpoBe,lJ;HTe
6HcceKTpHcy AM" H BbICOTY AH. KaxoH H3 OTpe3KOB HaH-
60JlbmHH AM", AH HJlH AB?
6. HaqepTHTe OCTpOyroJThHbIH MBC, AB = AC = 3 CM. IIpoBeAH-
Te MeAHaHy AM H BbICOTY AH. H HTe AJlHHY OTpe3Ka MH.
7. B MBC BbI6paHa TOQKa 0 TaK, qTO MOB paBeH baCOB,
OA = OC, LBOC = 120°. HaH,lJ;HTe LAOC.
BapuaKm 4
1. HaQepTHTe TpeyrOJILHHK, npOBe,lJ;HTe 6HcceKTpHcbI. Bcel' a
Jm To'tJ:Ka nepeceqeHHH 6HcceKTpHc HaxO,n;HTCH BO BHYTpeH-
HeH 06J1acTH TJ)eyrOJlhHHKa?
2. BepHo JIB YTBepmAeHHe: JII060H TpeyrOJIbHHK HMeeT TPH
Me HaHhI?
3. BepHO JIB YTBepJKAeHHe: B TynOYI'OJIhHOM TpeyrOJlLHHKe
TOQKa rrepece'tJeHHH npo.n;OJImeHHH BhICOT HaxOAHTCH BO
BHemHeH 06JIaCTH TpeyrOJIbHHKa?
4. MOjKeT JIH 6HcceKTpHca TpeyrOJIbHHKa COBllMaTL co CTOpO-
HOH TpeyroJlLHHKa?
5. HaqepTHTe OCTPOYI'OJILHLIH pa3HocTopOHHHH MBC. I1po-
Be.n;HTe 6HcceKTpHcy AN H BbICOTY AH. KaKoH H3 OTpe3KoB
HaHMeHbIDHH AN, AH HJIH AB?
6. HaqepTHTe ocTpoyrOJIbHbIH baCDE, CD = DE = 6 CM. ITpo-
Be.n;HTe BLICOTY DH H 6HcceKTpHCY DN. HaU,lJ;HTe )J;JlHHY OT-
pe3Ka HN.
7. B L\ABC BhI6paHa To'tJKa 0 TaK, qTO MOB paBeH COB,
OA = DC, LAOB = 130°. HaH.n;HTe LAOC.
38
CaMOCTOHTeJIbHaH pa60Ta NQ 18
BapuaH,m 1
1. B paBHo6e,n;peHHoM TpeyrOJIbHHKe OCHOBaHHe B 3 pa3a
MeHbllIe 60KOBOH CTOpOHbI, a rrepHMeTp paBeH 56 CM. HaH-
,n;HTe OCHOBaHHe.
2. IIepHMeTp paBHo6e,n;peHHoro BC C OCHOBaHHeM BC pa-
BeH 36 CM, a nepHMeTp paBHOCTopOHHero BCD paBeH
42 CM. HaH)];HTe AB.
3. B paBHo6e,n;peHHoM MEC C OCHOBaHHeM BC rrpOBe,n;eHa Me-
,n;HaHa AD. Haft)];HTe Me,n;HaHY AD, eCJIH nepHMeTp ABC
paBeH 44 CM, a rrepHMeTp MED paBeH 28 CM.
4. BepHo JIH YTBepm,n;eHHe: B paBHbIX TpeyrOJIbHHKaX Me,n;Ha-
HbI, rrpOBe,n;eHHbIe K paBHbIM CTopOHaM paBHbI?
5. BepHo JIH YTBepm,n;eHHe: B paBHo6e,n;peHHoM TpeyrOJIbHHKe
Bce yrJlbI paBHbI?
6. Me,n;HaHa AM MEC paBHa OTpe3KY BM. BepHo JIH YTBep-
m,n;eHHe: O,n;HH H3 yrJIOB MEC paBeH CYMMe ,n;BYX ,n;pyrHx
yrJIOB?
7. B PaBH06e,n;peHHOM TpeyrOJIbHHKe O,n;HH H3 yrJIOB Ha 30 rpa,n;y-
COB MeHbllIe )];pyroro. H Te yrJIbI TpeyrOJIbHHKa.
Bapuanm 2
1. B paBHo6e,n;peHHoM TpeyrOJIbHHKe OCHOBaHHe B 4 pa3a
MeHbme 60KOBOH CTOpOHbI, a rrepHMeTp paBeH 63 CM. HaH-
,n;HTe OCHOBaHHe.
2. TIepHMeTp paBH06e,n;peHHoro !1CDE C OCHOBaHHeM CE pa-
BeH 34 CM, a nepHMeTp paBHOCTopOHHero !1DEF paBeH 39
CM. Haft:,n;HTe CE.
3. B paBHo6e,n;peHHoM MEC C OCHOBaHHeM BC npOBe)];eHa Me-
,n;HaHa AN. HaH,n;HTe Me,n;HaHY AN, eCJIH nepHMeTp MEC
paBeH 40 CM, a nepHMeTp MEN paBeH 24 CM.
4. BepHo JIH YTBepm,n;eHHe: B paBHo6e.n;peHHoM TpeyrOJIbHHKe
yrJIbI IIPH OCHOBaHHH PaBHbI?
5. BepHo JIB YTBepm,n;eHHe: B paBHbIX TpeyrOJIbHHKaX Me,n;Ha-
HbI, npOBe,n;eHHbIe K paBHbIM CTopOHaM paBHbI?
6. Me,n;HaHa AN MEC paBHa OTpe3KY BN. BepHo JlH YTBep-
m,n;eHHe: O)];HH H3 yrJIOB MEC paBeH pa3HOCTH ,n;BYX ,n;pyrHx
yrJIOB?
7. B paBHo6e,n;peHHOM TpeyrOJIbHHKe O H H3 yrJIOB Ha 60 rpa,n;y-
COB MeHbllIe ,n;pyroro. Haii HTe yrJIbI TpeyrOJIbHHKa.
CaMOCTOHTeJIbHaH pa60Ta N2 18
39
Bapuanm 3
1. B paBHo6e,ZJ;peHHoM TpeyrOJIbHHKe OCHOBaHHe B 5 paa MeHb-
me 6oKoBoH CTOpOHbI, a nepHMeTp paBeH 55 CM. HaH,lJ;HTe
OCHOBaHHe.
2. IIepHMeTp paBHo6e,ZJ;peHHoro C C OCHOBaHHeM BC pa-
BeH 35 CM, a nepHMeTp paBHOCTopOHHero !1BCD paBeH 45
CM. HaH BTe AB.
3. B paBHo6e,ZJ;peHHoM !1CDE C OCHOBaHHeM DE npOBe eHa Me-
HaHa CM. Haii,D;HTe Me,lJ;HaHY CM, eCJIH nepHMeTp !1CDE
paBeH 48 CM, a nepHMeTp !1CDM paBeH 32 CM.
4. BepHo JIH YTBepm,ZJ;eHHe: B paBHOCTopOHHeM TpeyrOJIbHMKe
Bce yrJIbI paBHbI?
5. BepHo JIB YTBepm,ZJ;eHHe: B paBHo6e peHHOM TpeyrOJIbHMKe
6HcceKTpHca, npOBe,n;eHHa8 K OCHOBaHMIO, HBJIHeTCH Me-
.A H 8.HOH M BhICOTOH?
6. Me 8Ha AM C paBHa 0Tpe3KY BM. BePHO JIM yrBepm.n;e-
IDle: OAHH B3 yrJIOB C paBeH CYMMe BYX ,n;pyrMX yrJIOB?
7. B paBH06e,zq>eHHoM TpeyrOJIbHMKe O,n;HH H3 yrJIOB Ha 15 rpa-
.AY COB MeHhme ,lJ;pyroro. HaU,lJ;MTe yrJIbI TpeyrOJIbHHKa.
BapuaH,m 4
1. B paBHo6e.ApeHHOM TpeyrOJIbHBKe OCHOBaHMe B 2 pa3a
MeHbme 60KOBOH CTOpOHhI, a nepHMeTp paBeH 60 CM. Haii-
)J;HTe OCHOBaHHe.
2. IIepHMeTp paBHo6e peHHoro !1CDE C OCHOBaHHeM CE paBeH
30 CM, a rrepHMeTp paBHOCTopOHHero !1DEF paBeH 36 CM.
HaHAHTe CE.
3. B p8.BHo6e peHHoM t1CDE C OCHOBaHHeM DE npOBe,ZJ;eHa Me-
.AHaHa CN. HaH,D;MTe Me,lJ;MaHY CN, eCJIH nepHMeTp !1CDE
paBeH 42 CM, a rrepHMeTp !1CDM paBeH 26 CM.
4. BepHo JIB YTBepm eHHe: B paBHo6e,ZJ;peHHOM TpeyrOJIbHHKe
6HcceKTpHca, npOBe eHHaH K OCHOBaHHIO, HBJIHeTCH Me-
AuaHOH H BbICOTOii?
5. BepHO JIM YTBepm,D;eHMe: B paBHOCTopOHHeM TpeyrOJIbHMKe
Bce yrJIhI paBHbI?
6. MeAHaHa AM MBC paBHa OTpe3KY BM. BepHO Jm ymepjK)J;e-
HHe: OAHH M3 yrJIOB MBC p8BeH paaHOCTH ,D;BYX PyrHX yrJIOB?
7. B paBH06e,lJ;peHHoM TpeyrOJIbHHKe O HH H3 yrJIOB Ha 45 rpa-
,lJ;YCOB MeHbllIe ,D;pyroro. Haii,D;HTe yr JIbI TpeyrOJIbHHKa.
40
CaMOCTOHTeJIbHaSl pa60Ta N2 19
Bapuanm 1
1. OTpe3KH AB H CD rrepeceKalOTcH B TOqKe 0 - cepe HHe OT-
pe3Ka AB, LOAD paBeH LOBC. HaH HTe BC, eCJIH AD paB-
HO 12 CM.
2. Ha 6HcceKTpHce LA B3HTa TO'tlKa D, a Ha cTopoHax 3Toro yr-
JIa - TOqKH B H C TaKHe, 'tITO LADB paBeH LADC. CpaBHHTe
BD H CD.
3. BepHO JIH yrBepm.n;eHHe: B paBHblX TpeyroJIbH.HK8X 6HCceKTpH-
ChI, rrpoBe eHHhIe K COOTBeTCTBeHHO paBHbIM CTOpoHaM, paBHbI?
4. TIpSIMaSI, rrepneH,n;HKYJIHpHaH K 6HcceKTpHce LA, rrepeceKa-
eT CTOpOHhI yr JIa B TOqKaX M H N. HBJISIeTCH JIH TpeyrOJIbHHK
AMN paBHo6e,n;peHHhIM?
5. Ha CTopOHe AC KaK Ha OCHOBaHHH rrOCTpoeHbI no O HY CTOpO-
HY OT Hee ,n;Ba paBH06e peHHbIx TpeyrOJIbHHKa ABC H
MMC. BepHO JIH YTBepjK eHHe: rrpHMa.H BM rrepeceKaeT CTO-
POHY AC H eJIHT ee B OTHorneHHH 1 : 3?
6. BepHo JIH YTBepm,ZJ;eHHe: B paBHOCTopOHHeM TpeyrOJIbHHKe
Bce yr JIbl paBHhI?
7. BepHo JIH YTBepjK eHHe: eCJIH Me,ZJ;HaHa TpeyrOJIbHHKa COB-
rra,n;aeT C ero BhICOTOii, TO TpeyrOJIbHHK paBHOCTopOHHHH?
Bapuanm 2
1. OTpe3KH AB H CD rrepeceKalOTcH B TOQKe 0 - cepe HHe OT-
pe3Ka AB, LADO paBeH LOCB. HaH HTe AD, eCJIH BC paB-
HO 11 CM.
2. Ha 6HcceKTpHce B B3HTa TOqKa C, a Ha CTopOHax 3Toro yr-
JIa - TO KH D HE TaKHe, qTO LDCB paBeH LBCE. CpaBHHTe
BD H BE.
3. BepHo JIH YTBepm,n;eHHe: B paBHbIX oCTpoyrOJIbHbIX Tpe-
yrOJIbHHKax BhICOThI, IIpOBe eHHble K COOTBeTCTBeHHO paB-
HhIM CTopOHaM, paBHhI?
4. IIpHMaH, He nepneH HKYJIHpHaH K 6HcceKTpHce LB, nepece-
KaeT CTOpOHbI yrJIa B TO'tlKaX C H D. HBJIHeTCH JIH TpeyrOJIb-
HHK BCD paBHo6e,n;peHHbIM?
5. Ha CTopOHe AC RaE Ha OCHOBaHHH nOCTpoeHbI no O HY CTOpO-
HY OT Hee Ba paBH06e,n;peHHhIx TpeyrOJIbHHKa MBC H C.
BepHo JIH yrBepm,ZJ;eHHe: npHMaSI BN rrepeceKaeT CTOpoHY AC
H eJIHT ee B OTHomeHHH 1 : 21
6. BepHo JIH YTBepm,ZJ;eHHe: eCJIH Me,n;HaHa TpeyrOJIbHHKa COB-
na aeT C ero BbICOTOH, TO TpeyrOJIbHHK paBH06e,n;peHHhIH?
7. BepHo JIH YTBepm,n;eHHe: B paBHOCTopOHHeM TpeyrOJIbHHKe
Bce yrJIbI paBHbI?
CaMOCTOSlTeJIbHaH pa60Ta M 19
41
BapuaKm 3
1. OTpe3KH AB H CD nepeceK8lOTcH B TOqKe 0 - cepe HHe OT-
pe3Ka AB, LOAD paBeH LOBC. HaH,lI;HTe BC, eCJIH AD paB-
HO 14 CM.
2. Ha 6HcceKTpHce LA B3HTa TOqKa N, a Ha cTopoHax 3Toro yr-
Jla - TOqKH B H C TaKHe, qTO LANB paBeH LANC. CpaBHHTe
AB HAC.
3. BepHo JIB ymepm,n;eHHe: B paBHbIX 'rpeyroJlbHHKax 6HcceKTPH-
CDI, npoBe eHHhIe K COOTBeTCTBeBHO paBHhIM CTOpoHaM, paBHbI?
4. IIpHMaH, He rreprreHAHKYJlHpHaH K 6HcceKTpHce LA, rrepece-
KaeT CTOpOBbI yrJIa B TO"tlKaX M H N. HBJIHeTCH JlH TpeyrOJIh-
BBK AMN paBHo6e,n;peBHhIM?
5. Ha CTopoHe AC KaK Ha OCHOBaHHH nOCTpoeHhI no O HY CTOpo-
HY OT Hee ,lI;Ba paBHo6e peHHbIx TpeyroJlhHHKa ABC H
AAMC. BepHO JIB yTBepm):teHHe: rrpHMaSi BM rrepeceKaerr CTO-
POllY AC H eJlHT ee B OTHomeHHH 1 : 4?
6. BepHo JIB YTBep)K eHHe: B paBHo6e peHHOM TpeyrOJIhHHKe
Bce yrJlhI paBHhI?
7. BepHO JIll YTBepm eHHe: eCJlH Me HaHa TpeyroJlhHHKa COB-
rr aeT c ero BhICOTO:U, TO TpeyroJlhHHK paBHOCTopOHHHH?
BapuaH,m 4
1. OTpe3KH AB H CD nepeceKalOTcH B TO"tlKe 0 - cepe HHe OT-
pe3K8 AB, LADD paBeH LOCB. HaH HTe AD, eCJlH BC paB-
HO 15 CM.
2. Ha 6HcceKTpHce LB B3.HTa TO'tlKa C, a Ha CTopOHax 3Toro yr-
JIa - TOqKH D H E TaKHe, 'tITO LDCB paBeH LBCE. CpaBHHTe
DC B CE.
3. BepHo JIM YTBepm eHHe: B paBHhIx oCTpoyrOJIhHLIX Tpe-
yroJlbBHKax BhICOThI, npOBe eHHhIe K COOTBeTCTBeHHO paB-
HblM CTopOHaM, paBHbI?
4. TIp.HMa.H, nepneH,ZJ;HKYJlHpHaH K 6HcceKTpHce LB, nepeceKa-
eT CTOpOHhI yr JIa B TO'tlKaX C H D. .HBJIHeTC.H JlH TpeyroJlhHHK
BCD paBHo6e,n;peHHhIM?
5. Ha CTopoHe AC KaK Ha OCHOB8.HHH rrOCT})oeHhI no O,ZJ;HY CTOpo-
By OT HOO ,n;Ba p8.BHo6e peHHhIx TpeyrOJIhHHKa MBC H MNC.
BePHO JIM yrBepJK eHHe: rrpHMaH BN nepeceKaeT CTOPOHY AC
H .n;eJIHT ee B OTHomeHHH 1 : 5?
6. BepHo JIM YTBepm.n;eHHe: ecJIH 6HcceKTpHca 11JeyrOJIhHHKa COB-
n aeT C ero BhICOTOH, TO TJ)eyrOJIhHHK paBH06e peHHbIH?
7. BepHo JIH YTBepJK eHHe: B paBHo6e peHHoM TpeyroJlLHHKe
Bce yrJIbI paBHhI?
42
CaMOCTOHTeJlhHaH pa60Ta M 20
Bapualtm 1
1. BepHo JIM YTBepmAeHHe: eCJIH CTOpOHa O HOrO paBHOCTO-
pOHHerO TpeyrOJIbHHKa paBHa CTOpOHe APyroro paBHOCTO-
pOHHerO TpeyrOJIbHHKa, TO TpeyrOJIhHHKH paBHhI?
2. Ha OTpe3Ke AC KaK Ha OCHOBaHHH rrOCTpoeHhI no pa3HbIe
CTOpOHbI OT Hero Ba paBH06e.n;peHHhIX TpeyrOJIhHHKa MBC
H AADC. BepBO JlH yrBepm.n;eHHe: OTpe3KH AC H BD nepneH-
HKYJIHpHbI?
3. ,n:aH "tJeThIpexyroJlbHHK ABCD, AB = CD, BC = AD, AC -
AHarOHaJIb. CpaBHHTe LACB H LCAD.
4. OTpe30K npHMoH AB TO"tJKaMH P H Q AeJIHTCH Ha TPH paB-
Hble tlaCTH. BHe OTpe3Ka AB no O HY CTOPOHY OT Hero B3HTbI
TOqKH C H D TaK, '[{TO AC = BD H CQ = DP, CYMMa LDPB H
LCQA paBHa 140°. Hai1 HTe LCQA.
5. OTpe3KH AB H CD rrepeceKalOTCH B HX 06:w;eii cepe HHe -
TOqKe O. BepHo JIH YTBepmAeHHe: OTpe3KH AC H BD paBHhI?
6. YrJIhI TpeyrOJlhHHKa OTHOCHTCSI KaK 2 : 3 : 4. HaH HTe
MeHhlliHH yrOJI.
7. BepHo JlH YTBepjK eHHe: B paBH06e peHHoM TpeyroJlhHHKe
6HcceKTpHca, rrpOBetJ;eHHaH K OCHOBaHHIO, SlBJlHeTCH Me-
AHaHoH H BhICOTOH?
Bapualtm 2
1. BepHo JIH YTBepjKAeHHe: eCJIH CTopOHa OAHoro paBH06e,n;-
peHHoro TpeyrOJIhHHKa paBHa CTopOHe ,n;pyroro paBH06e -
peHHoro TpeyrOJIhHHKa, TO TpeyrOJIbHHKR paBHhI?
2. Ha OTpe3Ke BD KaK Ha OCHOBaHHH nOCTpoeHhI no pa3HbIe
CTOpOHhI OT Hero Ba paBHo6e.n;peHHbIX TpeyrOJIhHHKa
Ili3CD H ABDE. BepHo JlH YTBepm,n;eHHe: npHMhIe BD H CE
neprreHAHKY JlHPHhI?
3. ,D;aH tleThIpexyroJlbHHK ABCD, AB = CD, BC = AD, BD-
.n;HarOHaJIb. CpaBHHTe LCBD H LBDA.
4. OTpe30K npHMoH AB TO'[{KaMH N H M eJIHTCH Ha TpH paB-
HhIe qaCTH. BHe OTpe3Ka AB no O,ZJ;HY CTOPOHY OT Hero B3HTb!
TOtlKH C H D TaK, TO AC = BD H CM = DN, CYMMa LDBN H
LCAM paBHa 120°. HaiiAHTe LDBN.
5. OTpe3KH AB H CD nepeceKalOTCH B HX 06 eH cepe.n;HHe -
TOtlKe o. BepHo JIH YTBepmAeHHe: LOCA H LODB paBHLI?
6. YrJlhI TpeyrOJIbHHKa OTHOCHTCH KaK 1 : 2 : 3. HaH HTe 60JIh-
illHH yroJl.
7. BepHo JIH YTBepmAeHHe: B paBH06e.n;peHHoM TpeyrOJlhHHKe
yr JIbI npH OCHOBaHHH paBHbI?
CaMOCTOHTeJIbHaH pa60Ta N2 20
43
Bapuaum 3
1. BepHO JIH YTBepm eHHe: eCJIH )J;Be CTOpOHbI O,lJ;HOrO Tpe-
yrOJIhHHRa paBHbI BYM CTopOHaM ,n;pyroro TpeyrOJIbHHKa,
TO TaKHe TpeyrOJIbHHKH paBHhI?
2. Ha oTpeaKe AC KaK Ha OCHOBaHHH nOCTpoeHbI no pa3HbIe
CTOpoHhI OT Hero ,lJ;Ba paBHo6e.n;peHHbIX TpeyrOJIbHHKa MEC
H MDC. BepHo JIH YTBep eHHe: npHMbIe BD HAC nepneH-
.n;UKY JISlpHhl?
3. AaH eThIpexyrOJIhHHK BCDE, BC = DE, CD = BE, BD -
,lJ;HarOHaJlh. CpaBHHTe LCDB H LDBE.
4. OTpeaoK npHMoH AB TOqKaMH P H Q ,lJ;eJIHTCH Ha TPH paB-
Hble qaCTH. BHe OTpeaKa AB no O HY CTOPOHY OT Hero B3RThI
TO KH C H D TaK, qTO AC = BD H CQ = DP, CYMMa LACQ H
LPDB paBHa 150°. HaH,lJ;HTe LPDB.
5. Orpe3KH NM H PQ nepeceKMOTCH B HX o6:rn;eH cepe,n;HHe -
TOqKe o. BepHO JIH yTBepm,n;eHHe: OTpe3KH PN H MQ PaBHbI?
6. YrJIhI TpeyrOJIbHHKa OTHOCHTCH KaK 1 : 3 : 5. Haii,ZJ;HTe
MeHbmHH yrOJI.
7. BepHo JIH YTBepm eHHe: eCJIH TPH CTOpOHhI o,n;Horo Tpe-
yrOJIbHIIKa COOTBeTCTBeHHO paBHbI TpeM CTopOHaM ,u;pyroro
TpeyrOJIbHHKa, TO TaKHe TpeyrOJIbHHKH paBHbI?
Bapuanm 4
1. BepHo Jill YTBepm,lJ;eHHe: eCJIH TPH CTOpOHbI O,lJ;HOro Tpe-
yroJIhHHKa COOTBeTCTBeHHO paBHbI TpeM CTopOHaM .n;pyroro
TpeyrOJIbHHKa, TO TaKHe TpeyrOJIbHHKH paBHbI?
2. Ha OTpeaKe BD RaE Ha OCHOBaHHH nOCTpoeHbI no pa3HbIe
CTOpoHbI OT Hero Ba paBHo6e,n;peHHbIX TpeyrOJIbHHKa BCD
H BDE. BepHO JIH YTBepm,n;eHHe: OTpe3KH BD H CE nepneH-
.AHKY JIHPHbI?
3. ,lI;aH qeTbIpexyrOJIbHHK BCDE, BC = DE, CD = BE, CE -
.AHarOH8.JIb. CpaBHHTe LDCE H LBEC.
4. OTpe30K npHMoii AB TOqKaMH N H M .n;eJIHTCH Ha TPH paB-
Hble qaCTH. BHe OTpe3Ka AB no O,n;HY CTOPOHY OT Hero B3HTbI
TOqKH C H D TaR, qTO AC = BD II CM = DN, CYMMa LDNB II
LCMA paBHa 130°. HaH,lJ;HTe LDNB.
5. OTpe3KH NM II PQ nepeceKaIOTCH B HX o6m;ei1: cepe HHe -
TOqKe o. BepHo JIH YTBep:>K,lJ;eHHe LPNO H LQMO paBHbI?
6. 'YrJIbI TpeyrOJIbHHKa OTHOCHTCH KaK 2 : 4 : 3. Haii,lJ;HTe
60JIbIIIllii yrOJI.
7. BepHo JIH yrBepm,n;eHHe: B paBHOCTOpOHHeM TpeyrOJIbHHRe
Bce yrJIbI paBHbI?
44
CaMOCTOHTeJIbHaH pa60Ta NQ 21
Bapuanm 1
1. OTpe3KH CD H MN - ,ZJ;HaMeTpbI oKpymHocTH C eHTpOM O.
BY YT JIH paBHbI XOp,ZJ;bI CN H MD?
2. OTpe30K AB - HaMeTp oKpymHocTH C eHTpOM 0, AC Ii
BC - paBHhIe XOP,ZJ;hI 3TOH oKpymHoCTH. Haii,ZJ;HTe LAOC.
3. OTpe3KH MN H EF - paBHbIe XOp)J;bI oKpymHocTH C eH-
TpOM o. OP - rrepneH,ZJ;HKYJIHp K MN, OD - IIepneH HKY-
JIHp KEF. CpaBHHTe oTpe3KH OP HOD.
4. MomHo JIM Ha npHMOH OTMeTHTb ,ZJ;Be TO'tlKH TaR, 'tIT06bI pac-
CTOHHHe MeiK,n;y HHMH 6bIJIO B,ZJ;BOe 60JIbme ,ZJ;aHHOrO oTpe3Ka?
5. BepHo JIH YTBepm,ZJ;eHHe: xop,n;a - 3TO OTpeaOK, coe,ZJ;HHHIO-
IIJ;HH ,ZJ;Be TO'tlKH oKpymHocTH?
6. Ha oKpymHocTH C eHTpOM 0 nOCJIe,n;OBaTeJIhHO B3,HTbI TOq-
KH A, B, C, D, E TaR, 'tITO TO'tIKH A 11 E - KOHD;hI ,ZJ;HaMeTpa;
LAOC = LCOE, LAOB = 60 , a LDOE = 30°. CpaBHHTe OT-
pe3KH BD H AC.
7. MoryT JIH TpH OKpymHOCTH HMeTb poBHO 9 TOQeK nepeceqeHHH?
Bapuanm 2
1. OTpe3KH AB H CD - )J;HaMeTphI oKpymHocTH C n;eHTpOM O.
BY,ZJ;YT JIH paBHbI XOPAbI AC H BD?
2. OTpe30K MN - HaMeTp oKpymHocTH C eHTpOM 0, MP H
N P - paBHbIe XOp,lJ;bI 3TOH oKpymHoCTH. Haii,ZJ;HTe LO MP .
3. OTpe3KH AB H CD - paBHbIe XOp)J;bI oKpymHocTM C eH-
TpOM O. 0 E - neprreH)];HKY JIHp K AB , OF - nepneH)J;HKY JIHp
K CD. CpaBHHTe OTpe3KH AE H DF.
4. MomBo JIM Ha I1pHMOH OTMeTHTh ,ZJ;Be TO'tlKH TaR, 'tITo6bI pac-
CTOHHHe MeiK,l(y HHMH 6bIJIO B,ZJ;Boe MeHhme ,ZJ;aHHOrO oTpe3Ka?
5. BepHo JIM YTBepm)J;eHHe: pa,ZJ;HYC - 3TO OTpe30R, coe,ZJ;HHHIO-
IIJ;HH n.eHTp C KaKOH-JIH60 TO'tlKOH oRpymHocTH?
6. Ha oKpymHocTH C II.eHTpOM 0 nOCJIe,n;OBaTeJIhHO B3HTbI TOq-
KH B, C, D, E, F TaK, 'tITO TO'tlKH B H F - KOHII.bI )J;HaMeTpa;
LBOD = LDOF, LBOC = 30°, a LEOF = 60°. CpaBBMTe OT-
pe3KH BD H EF.
7. MorYT JIll TpH oKpymHocTII HMeTb pOBHO 10 TO'tIeK nepece-
qeHHfl?
CaMOCTOHTeJlbHan pa60Ta M 21
45
BapuaKm 3
1. OTpe3KIf CD H MN - HaMeTpbI oKpymHocTH C o;eHTpOM O.
BY,lQ7T JIH paBHbI XOp bI CM If ND?
2.0Tpe30K AB - HaMeTp oKpymHocTH C n;eHTpOM 0, AC H
BC - paBHhIe XOp bI STOM oKpymHocTH. HaH HTe LOBC.
3. OTpe3KH MN H EF - paBHhIe XOp bI OKpY.iKHOCTH C o;eH-
TpoM O. OP - neplIeH BKYJIHp K MN, OD - nepIIeH IIKY-
JI$lP KEF. CpaBHHTe oTpe3KII PN II ED.
4. MomHo JIll Ha rrpHMoH OTMeTIITb Be TO"llKIi TaR, lIT06bI pac-
CToHHHe Me y HHMB 6bIJIO BTpoe 60JIbme aHHOrO OTpe3Ka?
5. BepHo JIH YTBepm eHHe: ,l];HaMeTp - STO xop,l];a, npOXO ,H-
aH Qepe3 eHTp oKpymHocTH?
6. Ha OKpy.mHOCTH C o;eHTpoM 0 nOCJIe OBaTeJIbHO B3,HTbI TOlI-
KH A, B, C, D, E TaR, 'tITO TOqKH A BE - KOHD;bI ,l];HaMeTpa;
LAOC = LCOE, LAOB = 60°, a LDOE = 30°. CpaBHHTe OT-
pe3KII AB II CE.
7. Moryr JIB TpH OKPYJRHOCTB HMeTb pOBHO 7 TO"lleK rrepece'tJ:e-
HIIH?
BapuaH,m 4
1. OTpe3KH AB H CD - HaMeTpbI oKpymHocTH C n;eHTpOM O.
By YT JIB paBHbI XOp bI AD II BC?
2. OTpe30K MN - HaMeTp oKpymHocTH C n;eHTpoM 0, MP II
NP - paBHbIe XOp,l];LI aTOH oKpymHoCTH. H HTe LNOP.
3.0Tpe3KH AB II CD - paBHbIe XOp bI oKpymHocTH C n;eH-
TpOM O. OE - IIepIIeH IIKYJlHP K AB, OF - IIeprreH HKYJIHp
K CD. CpaBHIITe OTpe3KH OE H OF.
4. MomBo JIB Ha I1pSlMOH OTMeTHTb ,l];Be T01IKH TaK, 'tJ:To6bI pac-
CTOSlHHe Mem,n;y HHMM 6bLJIO BTpoe MeHbllIe aHHOrO OTpe3Ka?
5. BepHo JIM YTBepm eHHe: ra - STO qaCTb oKpymHocTH?
6. Ha oKpymHocTH C eHTpOM 0 IIOCJIe,ZJ;OBaTeJIbHO B3HTb! TO'tJ:-
KH B, C, D, E, F TaK, 'tJ:TO T01IKH B II F - KOH bI HaMeTpa;
LBOD = LDOF, LBOC = 30 c , a LEOF = 60°. CpaBHHTe OT-
pe3KH CE H DF.
7. MorYT JIB TPH OKPY.iKHOCTB HMeTb pOBHO 8 TOqeK rrepecelIe-
HHH?
46
CaMOCTOSlTeJIbHaSi pa60Ta N2 22
Bapuanm 1
1. OTpe3KH AB H CD - ,lJ;HaMeTpbI oKpymHocTH C n.eHTpoM O.
HaH,lJ;HTe nepHMeTp MOD, eCJIH H3BeCTHO, 'tITO CB = 18 CM,
AB = 24 CM.
2. Ha oKpymHocTH C n;eHTpOM 0 OTMeqeHbI TOtIl\.H A H B TaR,
qTO LAOB - npSlMoH. OTpe30K BC - HaMeTp oKpymHoCTH.
CpaBHMTe XOp,lJ;bI AB II AC.
3. Ha npHMOH ,ZJ;aHhI ,n;Be TOqKH A Ii B. MomHo JIM Ha npo,n;OJI-
.iKeHHM.rryqa BA OTJIOmHTb OTPe30K BC TaR, qT06bI BC = 5AB?
4. BepHo JIll YTBepm,l];eHHe: pa,l];HYC - 3TO xop a, npOXO Slm;aH
qepe3 n.eHTp oKpymHocTH?
5. ,D:aH OCTpbIH LABC H JIyq CD. MomHo JIH nocTpoHTb LDCE
TaR, "tJTo6bI LDCE 6bIJI paBeH YTpoeHHoMY LABC?
6. BepmHHbI paBHOCTopOHHero MBC JIemaT Ha oKpymHocTH,
BD - ,ZJ;HaMeTp aTOM oKpymHocTH. Onpe,lJ;eJIHTe BH,lJ; MCD.
7. BepHO JIM yrBepm,n;eHHe: eCJIH ,l];Be CTOpOHbI H yrOJI Me.iK,lJ;y
HHMH O Horo TpeyrOJIbHHKa COOTBeTCTBeHHO paBHbI BYM
CTopOHaM H yrJIY Mem,n;y HHMH ,n;pyroro TpeyrOJIbHHKa, TO
TaKHe TpeyrOJIbHHKH paBHbI?
Bapuanm 2
1. OTpe3RH BC H DE - ,lJ;HaMeTphI oRpymHocTH C n;eHTpoM O.
Haif,ZJ;HTe rrepHMeTp !1COD, eCJIH H3BeCTHO, qTO BC = 26 CM,
BE = 15 CM.
2. Ha oKpymHocTH C n;eHTpOM 0 OTMeqeHbI TOqKH M H N TaK,
qTO LMON - npHMoH. OTpe30K NP - ,l];HaMeTp oKpymHoc-
TH. CpaBHHTe XOP,ZJ;bI MN H MP.
3. Ha npHMoM ,ZJ;aHbI Be TOqKH C H D. MO.iKHO JIH Ha npo,n;OJIme-
HRH a CD OTJIO.mHTb OTpe30K CE TaK, QTo6bI CE = 3CD?
4. BepHo JIH YTBepm,lJ;eHHe: xop,n;a Bcer.n;a MeHbme PBAHyca?
5. AaH TYnOH LAOB H JIyq AD. MomHo JIH nOCTpOHTb LDAE
TaR, QTo6hI LDAE 6bIJI paBeH 'tIeTBepTH LABC?
6. BepmHHbI paBHOCTopOHHero CDE JIemaT Ha OKPY.iKHOCTII,
DE - .D;H8MeTp aTOM oKpymHocTH. Orrpe eJIHTe BH,lJ; DEF.
7. BepHo JIH YTBepm,lJ;eHHe: H3 TOqKH, He 411Jema eM Ha npH-
MOH, MomHO npOBeCTH rreprreH,ZJ;HKYJIHp K aTOM npHMoM, H
npHTOl'li TOJIbRO O,lJ;HH?
CaMOCTOSlTeJIbHaH pa60Ta NQ 22
47
BapuaHm 3
1. Orpe3RH AB H CD - HaMeTpbI oKpymHocTM C n;eHTpOM O.
HaHAMTe nepHMeTp tJ30C, eCJIH H3BeCTHO, "liTO AD = 16 CM,
AB = 28 CM.
2. Ha oKpymHocTH C n;eHTpOM 0 OTMe"tleHbI TO"tlKH A, B H C
TaR, "liTO BC - .n;HaMeTp OKPYJKHOCTH, AO-LBC. CpaBHMTe
XOp.n;hI CA H AB.
3. Ha IIpHMOH .n;aHbI ABe TOqKH M H N. MomHO JIH Ha rrpo OJIme-
HIDI.nyqa MN OTJIOJKMTb OTpe30K MP TaR, "tITOOD! MP = 2MN?
4. BepHO JIM YTBepm,n;eHHe: xop.n;a - 3TO OTpe30K, coe)J;HHHIO-
m;HH n.eHTp C KaKOH JIM60 TOqKOH oKpymHocTH?
5. .naB TYnOH LABC H JIy"tJ AD. MOJKHO JIH nOCTpOHTb LDAE
TaK, 'IIT06LI LDAE 6bIJI paBea nOJIOBHHe LABC?
6. BepIIIHHbI paBHOCTopOHHero AABC JIemaT Ha oKpymHocTH,
BD - ,n;HaMeTp 3TOH oKpymHoCTH. Onpe eJIHTe BM.n; AABD.
7. BepHO JIM YTBepm eHHe: B paBHo6eApeHHOM TpeyrOJIbHHKe
yrJIhI npH OCHOBaHMH paBHLI?
BapuaKm 4
1. Orpe3KH BC H DE - )J;HaMeTpbI oKpymHocTM C n;eHTpOM O.
Haii.n;MTe IIepHMeTp tJ30E, eCJIM H3BecTHO, qTO DE = 32 CM,
CD = 11 CM.
2. Ha oKpymHocTH C n.eHTpOM 0 OTMeqeHbI TO'tJ:KM M, N H P
TaR, 'liTO NP - .n;HaMeTp oKpymHocTH, MO.lNP. CpaBHHTe
XOp,n;bI PM Ii MN.
3. Ha npHMoH .n;aHbI ,n;Be TO"llKH A M B. MomHo JIM Ha npo.n;OJIme-
HMM JIY"lIaAB OTJIOJKHTb OTpe30K AC TaR, QTo6LI AC = 4AB?
4. BepBO JIM YTBepm eHMe: pa.n;MYC Bcer.n;a 60JILme XOptJ;bI?
5. ,lI;aH OCTpbIH LAOB H JI BC. MomHo JIH nOCTpOHTL LCBD
TaK, tlTo6LI LCBD 6hIJI paBeH y BoeHHOMY LAOB?
6. BepmHHbI paBHOCTopOHHe:ro CDE JIemaT Ha OKPYJKHOCTH,
DE - AHaMeTp aTOM OKPY.iKHOCTH. Onpe eJIHTe BH.n; CEF.
7. BepHO JIM YTBepmtJ;eHHe: B paBHo6e,n;peHHOM Tpey:rOJIbHMKe
6HcceKTpHca, rrpOBe,n;eHHaH K OCHOBaHHJO, HBJIHeTCH Me.n;H-
aHOH H BLICOTOii?
48
CaMOCTOHTeJlbHaH pa60Ta N 23
BapuaKm 1
1. aHbI oKpymHocTb, TOqKa A, He JIemam;aH Ha HeM, H oTpe-
aOR BC. Bcer.n;a JIH MomHO IIOCTpOHTb Ha oKpymHocTH TOq-
KY D TaR, QTo6bI AD = BC?
2. aH TyrroH LMON. MomHo JIH nOCTpOHTb JIY'tJ: OS TaR,
'tJ:T06bI LSOM H LSON 6hIJIH paBHhIMH TYIIbIMH yrJIaMH?
3. BepHo JIll YTBepm.n;eHHe: B IIOMOrn;blO HPRYJIH II JIHHeHKH
HeB03MO.iKHO IIOCTpOHTb rreprreH)J;HKY JIHpHble IIpHMhIe?
4. MOjKHO JIH IIOCTpOIlTb C IIOMOrn;blO D;HPKYJIH H JIHHeHKH
paBHo6e.n;peHHbIH TpeyrOJIbHHK no 60KOBOH CTopOHe H yrJIY
IIPH BepmHHe?
5. IIocTpoHTb Tynoy:rOJIbHbIH pasHocToPOHHHH NP. Mom-
HO JIll IIOCTpOHTb oKpymHocTb, npOXO)J;Hm;ylO qepea TO'tJ:KII
M, N II P?
6. MO.iKHO JIH C rroMOID;blO HPRYJIH H JIHHeHKH IIOCTpOHTb
yrOJI B 45°?
7. OCHoBaHHe paBH06e.n;peHHoro MBC paBHO 8 CM. Me.n;HaHa,
npOBe,ll;eHHaH R 60KOBOH CTopOHe, pas6HBaeT TpeyrOJIbHHK
Ha .n;Ba TpeyrOJIbHHKa TaK, qTO IIepHMeTp O)J;Horo Tpey:rOJIb-
HHRa Ha 2 CM 60JIbme IIepHMeTpa .n;pyroro. HaH.n;HTe 60-
KOBYIO CTOPOHY /j. ABC.
Bapuanm 2
1. )J;aHbI oKpymHocTb, TO'tJ:Ra B, He JIejKaID;aH Ha HeH, H OTpe-
30K CD. Bcer.n;a JIll MomHO IIOCTpOHTb Ha OKPYjKHOCTH TO'tJ:-
KY E TaK, 'tJ:T06bI BE = CD?
2. AaH OCTpbIM LAOB. MO.iKHO JIM IIOCTpOIlTb JIY'tJ: OC TaR,
'tJ:To6bI LAOe H LBOe 6bIJIH paBHbIMH TYIIhIMH yrJIaMH?
3. BepHo JIll YTBep.iK.n;eHHe: B IIOMOrn;blO HPKYJIH II HHeHKH
HeB03MO.iKHO paa.n;eJIHTb OTpeaOK Ha .n;Be paBHbIe qaCTH?
4. MO:>KHO JIH IIOCTpOHTb C IIOMOID;bIO IJ;HPKYJIH H JIHHeHKH
paBHocTopOHHHH TpeyrOJIbHIIK no cTopoHe?
5. IIocTpoHTb IIpHMoyrOJIbHbIH paBHo6e.n;peHHhIH ABC.
MomHo JIH IIOCTpOHTb oKpymHocTb, npoxo.n;HID;YIO 'tJ:epea
TO'tJ:KH A, B H C?
6. MO.iKHO JIH C IIOMOIIJ;hlO HPKYJIH H JIHHeHKII nOCTpOHTb
yrOJI B 135°?
7. OCHoBaHHe paBHo6e.n;peHHoro NP paBHo 10 CM. Me,n;Ha-
Ha, npOBe.n;eHHaH K 60KOBOH CTopOHe, paa6HBaeT TpeyrOJIb-
HHK Ha ,n:Ba TpeyrOJIbHHKa TaR, 'tJ:TO rrepHMeTp O)J;Horo Tpey-
rOJIbHHKa Ha 4 CM 60JIbme IIepHMeTpa )J;pyroro. HaH,lI;HTe
60KOBYlO CTOPOHY NP.
CaMOCTOHTeJIhHaH pa60Ta N 23
49
Bapuanm 3 u
1. ,IJ;aHbI oKpymHoCTb, TOqKa C, He JIe.iKa:rn;aH Ha HeM, H OTpe-
30K DE. Bcer,n;a JIH MOjKHO IIOCTpOHTb Ha oKpymHocTH TO-
qKY F TaR, QTo6hI CF = DE?
2. )];aH TynoH LMON. MomHo JIH nOCTpOHTh JIy-q OP TaK,
QTo6b1 LPOM Ii LPON 6hIJIH paBHbIMH TynbIMIi yrJIaMH?
3. BepHo JlH YTBepm eHHe: B nOMOID;bIO HPKYJIH 11 JIHHeHRH
HeB03MOjKHO nOCTpOHTb yroJl, paBHbIH ,ZJ;aHHoMY?
4. MomHO JlH nOCTpOHTb C nOMOnJ;blO IJ;HPKYJIH H JIHHeHKH
npHMoyrOJIhHhIH paBHo6e peHHbIH TpeyrOJIhHHK?
5. IIOCTpOHTb oCTpoyrOJIbHhIH paBHo6e peHHbIH MNP.
MomHo JIM nOCTpOHTb oKpymHocTb, IIpOXO.n;j{ YIO 'tIepea
TOqKH M, N H P?
6. MomHO JIH C TIOMO blO HPKYJISI H JIHHeHKH IIOCTpOHTb
yroJl B 22,5°?
7. OCHOBaHHe paBHo6e.zu>eHHoro MBC paBHO 6 CM. Me,lJ;HaHa,
npOBe eHHaH K 60KOBOH CTopOHe, paa6HBaeT TpeyroJlhHHK
Ha Ba TpeyrOJIbHHKa TaR, qTO nepHMeTp O Horo TpeyrOJIb-
HHKa Ha 2 CM 60Jlbme nepHMeTpa pyroro. HaH,lJ;MTe 60RO-
BYIO CTOPOHY MBC.
Bapuanm 4
1. AaHLI oKpymHocTh, TO Ka D, He JIemarn;aH Ha HeH, H OTpe-
30K AB. Bcer,ZJ;a JIH MomHO IIOCTpOHTh Ha OKPY.iKHOCTH TO"tJ-
KY C TaR, "tJTo6hI DC = AB?
2. aH OCTpbIH LAOB. MomHo JIH IIOCTpOHTh JIyq ON TaR,
'tIT06hI LAON H LBON 6bIJIH paBHbIMH TYIIbIMH yrJIaMH?
3. BepHo JIH YTBepm eHHe: B nOMOIIJ;blO II.HPKYJIH H JIHHeHKH
HeB03MomHO nOCTpOHTh 6HcceRTpHcy yrJIa?
4. MOJRHO JlH nOCTpOHTb C nOMOnJ;blO HPKYJIH Ii JIHHeHRH
KB paT no cTopoHe?
5. TIOCTpoHTh npHMoyrOJIbHhIH pa3HocToPOHHHH MBC Mom-
HO JIH nocTpoHTh oKpymHocTh, IIpOXO,lJ;H:rn;yIO "tJepe3 TO"tJKH
A, B H C?
6. MomHo JIB C IIOMOID;blO IJ;HPRYJIH H JIHHeHRH nOCTpoliTb
yrOJI B 112,5°?
7. OCHoBaHHe paBH06e,lJ;peHHoro f1MNP paBHO 12 CM. Me.n;Ha-
Ha, npOBe.n;eHHaH K 60KOBOH CTopOHe, pa36HBaeT TpeyrOJIb-
HHK Ha ,lJ;Ba TpeyrOJIbHHKa TaK, "tJTO rrepHMeTp O,lJ;HOrO Tpey-
rOJIbHHKa Ha 4 CM 60JIbme rrepHMeTpa ,lJ;pyroro. Haii,lJ;HTe
60KOBYlO CTOPOHY f1MNP.
50
CaMOCTOHTeJIbHaH pa60Ta N2 24
BapuaKm 1
1. ,l];aH 'tJ:eTLIpexyrOJIbHIIK ABCD, TOqKa M JIemHT Ha BC, a
TOqKaN - HaAD. OTpe3KHBMHNDnapaJIJIeJILHbI. By,nYT
JIH napaJIJIeJILHbI OTpe3KR MC H AN?
2. ,n:aH C, AD H CE - Me,nHaHbI C. TOQKa 0 - To'tJ:Ka
rrepeCeQeHHH Me,nHaH. IIapaJIJIeJIbHbI JIH OTpe3KH DE II BD?
3. ,n:aH C, AB = BC. LBCE - BHeIlIHHH yrOJI TpeyrOJIbHH-
Ka, CD - 6HcceKTpHca LBCE, LBAC = 60 0 . IIapaJIJIeJIbHbI
JlH npHMbIe AB H CD?
4. ,n:aH qeTbIpexyrOJIbHHK ABCD, AB = CD, BC = AD. IIapaJI-
JIeJIbHbI JIH rrpHMbIe BC H AD?
5. BepHo JIH YTBepm eHHe: ,nBa paBHo6e,npeHHbIx TpeyrOJIbHHKa
paBHbI, ecJIH 6oKoBaH CTopOHa o,nHoro TpeyrOJIbHHKa paBHa
60KOBOH CTopoHe ,npyroro TpeyrOJIbHHKa?
6. B ABC LA=38°, LB = 110°, LC = 32°. Ha CTopOHe AC
OTMeqeHbI TOqKR D H E TaK, qTO TOqKa D JIemHT Ha OTpe3Ke
AE, BD = DA, BE = EC. Hali,nHTe LDBE.
7. TIpHMhIe a, b H C - nonapHO rrepeceKaIOIqHecH. IIpsIMaH d naparr-
JIeJThHa IIpJIMOH b II He rrpoXO,AHT qepea TO'tJ:KY nepeceqeHHH IIpH-
MbIX a If C. Ha CKOJIbKO qacreH ,lI;eJurr ILJIOCKOCTb e?
Bapualtm 2
1. ,II:aH qeTbIpexyrOJIbHHK ABCD, TOqKa E JlemHT Ha AB, a
To'tJ:Ka F - Ha CD. Orpe3KH AE H CF rrap8JIJIeJIbHbI. By,ll;YT JIM
nap3JIJIeJIbHbI OTpe3KH BE H DF?
2. .D:aH C, AD H CE - Me,nH8.HbI C. To'tJ:Ka 0 - TOqKa
nepeCeqeHMH Me,nHaH. IIapaJIJIeJIbHbI JIH OTpe3KH OC II BD?
3. ,lJ:aH M3CD, BC = CD. LCDE - BHeIlIHHH yrOJI TpeyroJlbHH-
Ka, DF - 6HcceKTpHca LCDE, LCBD = 40°. IIapaJIJIeJIbHbI
.1IH npHMbIe BC H DF?
4. ,lJ:aH qeTbIpexyrOJIbHHK KLMN, KL = MN, LM = KN.
IIapaJIJIeJIbHbI JIH 'rrpHMbIe LM H KN?
5. BepHo JIH YTBepm,neHHe: Ba paBHOCTOpOHHHX TpeyrOJIbHH-
Ka paBHbI, eCJIH CTopOHa O Horo TpeyrOJIbHHKa paBHa CTO-
pOHe ,lI;pyroro TpeyrOJIbHHKa?
6. B ABC LA = 36 v , LB = 120°, LC=24°. Ha CTopOHe AC
OTMeqeHbI TOqKH D II E TaK, "liTO TOqKa D JIemHT Ha OTpe3Ke
AE, BD=DA, BE=EC. HaH,lI;HTe LDBE.
7. IIpslMhle a, b He - rronapHO nepeceKaIO ecH. IIpHMaH d ITapaJI-
JIeJlhHa npHMOH b H npOXO,ll;HT Qepe3 TOqKY nepeCeqeHIIH npH-
MbIX a Ii C. Ha CKOJILKO 'tJ:aCTeH ,lI;eJIHT nJIOCKOCTb 3TH rrpHMble?
CaMOCTOHTeJIbHaH pa60Ta N 24
51
Bapuaum 3
1. AaH qeTbIpexyrOJIbHHK ABCD, TOtIKa P JIemHT Ha BC, a
TOtIKa T - Ha AD. OTpe3KH BP II TD rrapaJIJIeJIbHbI. BY YT
JIH rrapaJIJIeJIbHbI OTpe3KH PC If AT?
2.,l];aH MEC, AD II CE - Me)J;HaHbI C. TOQKa 0 - TOqKa
nepeceqeHHH Me lIaH. IIapaJIJIeJIhHbI JIll OTpeaKH AO H BE?
3. AaH f1BCD, BC = CD. LCDE - BHellIHHH yrOJI TpeyrOJIbHH-
Ka, DF - 6HcceKTpnca LCDE, LCBD = 60°. IIapa.rrJIeJIbHbI
JIll npHMhle BC If DF?
4. ,l];aH qeTblpexyrOJIbHHK ABCD, AB = CD, BC = AD. IIapaJI-
JIeJIbHbI JIll rrpHMbIe AB II CD?
5. BepHO JIM YTBep eHlle: )J;Ba KBa,n;paTa paBHbI, eCJIII CTopoHa
O)J;Horo KBa,n;paTa paBHa CTopOHe pyroro KBa,n;paTa?
6. B C LA=43°, LB=90°, LC=27 c . HacTopoHeACoTMeqe-
HbI TO"tlKII D II E TaK, "tJ:TO TO\IKa D JIemliT Ha OTpe3Ke AE,
BD = DA, BE = EC. Haii HTe LDBE.
7. IIpHMble m, n H p - rronapHO nepeceKaro HecH npHMhIe.
IIpHMaH k napaJIJIeJIbHa rrpSIMoH n H He npOXO)J;HT qepea
TOqKY nepeCeqeHII.H rrpHMbIX m H p. Ha CKOJIbKO qaCTeH
)J;eJIHT rrJIOCKOCTb 3TH rrpHMbIe?
Bapuaum 4
i. ,1]:aH "tJ:eTbIpexyrOJIbHIIK ABCD, TOqKa M JIemHT Ha AB, a
TOqKa N - Ha CD. OTpeaKH AM Ii CN rrapaJIJIeJIbHbI. By,n;YT
JIll rrapaJIJIeJIbHbI OTpe3KH BM H DN?
2.,n:aH MEC, AD Ii CE - Me)J;HaHbI C. TOqKa 0 - TOqKa
nepece'tJeHHH Me,l);HaH. IIapaJIJIeJIbHhI JIM OTpe3KH OD H BE?
3. ,l];aH MBC, AB = BC. LBCE - BHemHliH yrOJI TpeyrOJIb-
HHKa, CD - 6HcceKTplica LBCE, LBAC = 50°. IIap8JIJIeJIb-
HbI JIll np.HMble AB H CD?
4. ,n:aH qeTblpexyrOJIbHHK KLMN, KL = MN, LM = KN.
IIapaJIJIeJIbHbI JIH rrp.HMble KL II MN?
5. BepHo JIH YTBepiK,l);eHlle: )J;Ba paBHo6e)J;peHHblx TpeyrOJIb-
HIIKa paBHbI, eCJIIi OCHOBaHlie O,l);HorO TpeyrOJIbHIIKa paBHO
OCHOBaHIIIQ ,n;pyroro TpeyrOJIbHIIKa?
6. B MEC LA = 37°, LB = 110°, LC = 43°. Ha CTopOHe AC
OTMeqeHbI TOqKH D II E TaK, qTO TOqKa D JIeiKIIT Ha OTpe3Ke
AE, BD = DA, BE = EC. HaH IITe LDBE.
7. IIpHMbIe m, n II p - rrorrapHO nepeceKaroIll;lIecH np.HMbIe.
IIpHMaH k rrapaJIJIeJIbHa npHMoH n Ii rrpOXO,l);HT qepea TO'tJKY
rrepeCeqeHIIH rrpSIMbIX m II p. Ha CKOJIbKO qaCTeH ,l);eJIHT
rrJIOCKOCTb 3TH rrpHMble?
52
CaMOCTOHTeJIbHaH pa60Ta N2 25
BapuaKm 1
1. OTpeaKIi AB M MN nepeceKaIOTCH B MX o6 eH cepe MHe.
BY YT JIM npHMble AM Ii BN napaJIJIeJIbHbI?
2. )];aH BblnYKJIbIH "tIeTblpexyrOJIbHHK MNPT, MN = NP,
LN M P paBeH LP MT. BepHo JIIi YTBepjK eHHe: npHMbIe N P
H MT napaJIJIeJIbHbI?
3. )];aH MBC, AB = BC, TO"tlKH D HE JIeiKaT Ha cTopoHaX AB Ii
BC cooTBeTcTBeHHo, LBCA = 80°, LEAC = 40°, AD = DE.
HaH liTe LBDE.
4. BepHo JIM yrBepm eHHe: eCJIH npli nepeCe"tleHMH BYX npH-
MbIX ceKyrn;e:ti HaKpeCT JIemarn;Me yrJIbI paBHbI, TO npHMhIe
rra paJIJIeJIbHbI?
5. )];aH AABC, AB = BC, LA = 30°, LBCE - BHellIHHH yrOJI
TpeyrOJIbHHKa. JI CD ,ZJ;eJIHT LBCE Ha Ba yrJIa, LDCE B
4 pa3a MeHbme, "tIeM LBCD. BepHo JIIi YTBepm eHHe: npH-
MLIe AB Ii CD napaJIJIeJIbHhI?
6. O HH li3 yrJIOB TpeyroJILHMKa paBeH 70°. Hai1,ZJ;liTe yrOJI Mem-
lIY 6HCCeKTPHCaMH BYX pyrHX yrJIOB aToro TpeyroJIbIrnKa.
7 )];aH MEC, LA = LC, AC : AB = 3 : 4; nepliMeTp AABC
paBeH 55 CM. HaH,ZJ;liTe AC.
Bapuanm 2
1.0TpeaKH CD H EF nepeceKaIOTCH B HX 06rn;eH cepe,ZJ;liHe.
By yr JIH rrpHMbIe CE M DF napaJIJIeJIbHbI?
2. )];aH BhInYKJIbIH QeTblpexyrOJIbHI1K ABCD, AB = BC,
LBAC paBeH LCAD. BepHo JIH YTBepm,ZJ;eHHe: npHMble BC
Ii AD rrapaJIJIeJIbHbI?
3. )];aH AABC, AB = BC, TO"tlKH D HE JIemaT Ha CTOpOHax AB H
BC cooTBeTcTBeHHo, LBCA = 80°, LEAC = 40°, AD = DE.
CpaBHHTe OTpe3KH BD M" BE.
4. BepHo JIH YTBepm)J;eHHe: eCJIH rrpli nepeCe"tleHHH BYX
npHMbIX ceKyrn;efl COOTBeTCTBeHHbIe yrJIhI paBHbI, TO
npHMble napaJIJIeJIbHbI?
5. )];aH ABC, AB = BC, LA = 40°, LBCE - BHellIHI1H yrOJI
TpeyroJIbHMKa. JIyq CD )J;eJIHT LBCE Ha Ba yrJla, LDCE B 6
paa MeHbIlIe, tIeM LBCD. BepHo JIM YTBep.iK eHHe: rrpHMbIe
AB H CD napaJIJIeJIbHbI?
6. O HH H3 yrJIOB TpeyrOJIbHHKa paaeH 60°. HaH)J;HTe yrOJI MeiK-
y 6liCCeKTpHCaMH BYX pyrMx yrJIOB 3Toro 1'peyrOJIbHHKa.
7. )];aH MBC, LA = LB, AB : BC = 3 : 5; nepHMeTp ABC
paBeH 52 CM. HaH HTe AB.
CaMOCTOHTeJIbHaH pa60Ta N2 25
53
Bapuanm 3
1. OTpe3KII AB II PS nepeceKaIOTCH B HX o6m;eii cepe,n;IIHe.
BY,lJ;YT JIH npHMbIe AP H BS napaJIJIeJIbHbI?
2. ,n:aH BblnYKJIhIH 'tIeTblpexyrOJIbHHK MNPT, MT = TP,
LNMP paBeH LPMT. BepHo JIH YTBepm,n;eHlIe: npHMhle
MN Ii PT rrapaJIJIeJIbHbI?
3. ,1];aH MBC, AB = BC, TOqKH D II E JIemaT Ha CTopOHax AB II
BC COOTBeTCTBeHHO, LBCA = 60°, LEAC = 30°, AD = DE.
Haii,D:IITe LBD E .
4. BepHo JIH YTBepm,lJ;eHHe: eCJIH npH nepeCe'tleHHH ,lJ;BYX npH-
MbIX ceKyw;eH CYMMa O)J;HOCTOpOHHHX yrJIOB paBHa 180°, TO
rrpHMbIe napaJIJIeJIbHbI?
5. ,1];aH MBC, AB = BC, LA = 20°, LBCE - BHemHHH yrOJI
TpeyrOJIbHHKa. JIY"tJ CD ,lJ;eJIIiT LBCE Ha ,l];Ba yrJIa, LDCE B
7 paa MeHbIIIe, qeM LBCD. BepHo JIH YTBepm,n;eHHe: IIPH-
MbIe AB II CD napaJIJIeJIbHhI?
6. O.n;HH H3 yrJIOB TpeyroJIbHHKa palleH 80°. H HTe yrOJI Mem-
,lJ;y 6HCCeKTpliCaMH BYX ,lJ;pyrnx yrJIOB 3Toro TIJeyrOJIhHHKa.
7. ,D;aH MBC, LA = LC, AC : AB = 4 : 5; rrepliMeTp MBC
paBeH 56 CM. HaH,lJ;HTe AC.
Bapuanm 4
1.0Tpe3KH CD H MP nepeceKaIOTcH B HX o6 eH cepe,n;HHe.
By,n;YT JIll npHMbIe CM H DP napaJIJIeJIbHbI?
2. ,1];aH BbInYKJIbIH "tJeTblpexyrOJIbHHK ABCD, AD = DC,
LBAC paBeH LCAD. BepHo JIll YTBepm.n;eHHe: npHMbIe AB
II CD napaJIJIeJIbHbI?
3. ,D;aH MBC, AB = BC, TOqKH D II E JIemaT Ha CTopOHax AB II
BC COOTBeTCTBeHHO, LBCA = 60°, LEAC = 30°, AD = DE.
CpaBHIITe OTpe3KII BD II BE.
4. BepHo JIH yrBepm)J;eHHe: rrpHMaH C Ha3bIBaeTCH ceKym;eii no
OTHomeHHIO K npHMbIM a II b, eCJIH OHa nepeceKaeT HX B
,lJ;BYX TOqKax?
5. ,n:aH MBC, AB = BC, LA = 50°, LBCE - BHemHHH yrOJI
TpeyrOJIbHHKa. JIyq CD AeJIRT LECE Ha ABa yrJIa, LDCE B
4 pa3a MeHbme, qeM LBCD. BepHo JIll YTBepm,lJ;eHlle: npH-
MhIe AB II CD napaJIJIeJIbHbI?
6. O,lJ;HH H3 yrJIOB TpeyrOJIbHHKa paBeH 50°. Ha Te yroJI Mem-
,n;y 6HcceKTpHCaMli ,n;BYX .n;pyrHX yr JIOB 3Toro TpeyrOJIbHHKa.
7. ,D;aH MBC, LA = LB, AB: BC = 2 : 7; rrepHMeTp MBC pa-
BeH 48 CM. HaH,lJ;HTe AB.
54
CaMOCTOSlTeJIbHaH pa60Ta N2 26
BapuaHm 1
1. B AABC LA = 40°, LBCE CMemHbIH C LACB, LBCE = 80°.
ITapaJIJIeJIbHa JIH 6HcceKTpHca LBCE rrpHMoH AB?
2. B BCD LB = 30°, LC = 70°. "l.{epea BepllIHHY C rrpOBe eHa
rrpHMaH CE TaK, qTO JIyq CD - 6HcceKTpHca LBCE. BepHo
JIH YTBepjK eHHe: rrpHMbIe BD H CE nap8.JIJIeJIbHbI?
3. ,[(aH AABC, BD - 6HcceKTpHca H BbICOTa. AC - 6HcceKTpH-
ca LBAE. BepHo JIll YTBepm eHHe: rrpHMbIe BC H AE
rrapaJIJIeJIbHbI?
4. AaH AABC, BD - 6HcceKTpHca, AD = BD, LC B .D;Ba paaa
60JIbIIIe LA. HaH,lJ;HTe LC.
5. BHcceKTpHcbI AK II CD paBHOCTopOHHero AABC nepeceKa-
IOTCH B TOtlXe O. HaH HTe LDOA.
6. ,IJ;aH paBHo6e.n;peHHbIH AABC c OCHOBaHHeM AC, AD - 6HC-
ceKTpllca, LBAD B 5 pa3 MeHbIIIe, tleM LADC. HaH IITe LB.
7. 113 BepmHHbI rrpHMOrO LC ABC rrpOBe.n;eHa BbICOTa CD.
Ha HTe LBCD, eCJIH LA = 40°.
Bapuanm 2
1. B M3CD LB = 30°, LCDE cMemHbIH C LBDC, LCDE =70°.
IIapaJIJleJIbHa JIH 6HcceKTplica LCDE rrpHMoH BC?
2. B MBC LA = 20°, LB = 80°. qepea BepmHHY B rrpOBe.n;eHa
npHMaH BD TaR, qTO Jlyq BC - 611cceKTpHca LABD. BepHO
JIH YTBepm eHlle: npHMble AC H BD napaJIJIeJIbHbI?
3. )];aH MNP, NS - 611cceKTpHca H BbICOTa. MP - 6HcceK-
TpHca LNMT. BepHo JIll YTBepm.n;eHHe: rrpHMbIe NP II MT
na paJIJIeJIbHbI?
4. ,[(aH AABC, BD - 6l1cceRTpllca, AD = BD, LC B meCTb pa3
60JIbme LA. HaH.n;HTe LC.
5. BucceKTpHcbI AN II CD paBHocTopoHHero AABC rrepeceKa-
IOTCH B TOtiKe O. HaH,lJ;IITe LCON.
6. AaH paBHo6e peHHbIH BCD c OCHOBaHHeM BD, DE - 6HC-
ceKTpHca, LBED B 5 pa3 60JIbme, qeM LCDE. HaH HTe LC.
7.113 BepmHHbI npHMoro LC ABC rrpoBe.n;eHa BbICOTa CD.
HaH.n;HTe LACD, eCJIH LB = 50°.
CaMOCTOHTeJIbHaH pa60Ta M 26
55
BapuaKm 3
1. B AABC LA = 50°, LBCE cMemHbIH C LACB, LBCE = 100°.
IIapaJIJIeJIbHa JIH 6HcceKTplica LBCE rrpHMoH AB?
2. B BCD LB = 50°, LC = 70°. qepea BepmHHY C rrpOBe eHa
npHMaH CE TaK, qTO JIytI CD - 6HcceKTpHca LBCE. BepHo
JIH YTBepm,ZJ;eHHe: rrpHMbIe BD H CE rrapaJIJIeJIbHbI?
3. ,lJ;aH AABC, BD - 6HcceKTpHca Ii BbICOTa. AC - 6HcceKTpH-
ca LBAE. BepHO JIM YTBepm eHHe: rrpHMhIe BD H AE
rrapaJIJIeJIbHhI?
4. ,n:aH AABC, BD - 6HcceKTpHca, AD = BD, LC B Ba pa3a
60JIbIIIe LA. Haii.D;HTe LA.
5. BHcceKTpHcbI AE II CD paBHocTopoHHero MBC rrepeceKa-
IOTCH B TOlIKe O. Haii,ZJ;HTe LDOE.
6.,L(aH paBHo6e,ZJ;peHHbIH MBC C OCHOBaHHeM AC, AD - 6HC-
ceKTp:nca, LBAD B 5 paa MeHbme, qeM LADC. HaH HTe LC.
7. Ha BepIIIHHhI rrpHMoro LC ABC rrpOBe,ZJ;eHa BLICOTa CD.
Haii MTe LBCD, ecJIH LA = 30°.
BapuaKm 4
1. B BCD LB = 35°, LCDE CMeiKHhIH c LBDC, LCDE = 80°.
IIapaJIJIeJIbHa JIH 6HcceKTpHca LCDE rrpHMoH BC?
2. B MBC LA = 40°, LB = 70°. YJepea BepmHHY B npOBe eHa
np.HMaH BD TaK, lITO JIY1I BC - 6HcceKTpHca LABD. BepHO
JIH YTBepm,ZJ;eHHe: rrpSiMble AC H BD rrapaJIJIeJIbHhI?
3. ,lJ;aH MNP, NS - 6HcceKTpHca II BbICOTa. MP - 6HcceK-
TpHca LNMT. BepHo JIH YTBepjK)J;eHHe: rrpHMhIe NS H MT
rrapaJIJIeJIbHbI?
4. ,n:aH MBC, BD - 6HcceKTpHca, AD = BD, LC B meCTb pa3
60JIbIIIe LA. HaH HTe LA.
5. BHcceKTpHcbI AM H CD paBHOCTopOHHero AABC rrepeceKa-
IOTCH B TOtlKe o. Haii,ZJ;HTe LAOC.
6. aH paBHo6e peHHbIH I1BCD c OCHOBaHHeM BD, DE - 6HC-
ceKTpHca, LBED B 5 paa 60JIbIIIe, tleM LCDE. HaH,ZJ;HTe LB.
7. Ha BepmHHbI np.HMOrO LC MBC rrpOBe eHa BhICOTa CD.
HaH HTe LACD, eCJIH LB = 60°.
56
CaMOCTOSlTeJIbHaH pa60Ta N 27
Bapuanm 1
1. TOqKa D - cepe HHa CTOpOHbI AB MEC. Qepe3 TOqKH B H
D IIpOBe eHbI rrpHMbIe b H d, rrapaJIJleJlbHbIe npHMoH AC.
IIepeCeKalOTCH JIB npHMbIe b H d?
2. qepea TOqKY, He Jlemam;ylO Ha npHMOH m, npOBe)J;eHbI TPH
rrpHMble. CKOJIbKO H3 aTHX npHMbIX rrepeceKaIOT npHMYIO m?
3. CYMMa HaKpeCT JIemam;HX yrJloB rrpH rrepeCeqeHHH BYX
napaJIJIeJlbHhIX rrpHMbIX ceKy eH paBHa 220 0 . HaH)J;HTe 3TH
yr JIbI.
4. HaH HTe Bce yrJIbI, o6paaoBaHHbIe npH rrepeCeqeHHH BYX
npHMbIX a H b ceKym;eH C, eCJIH O)J;HH H3 yr JIOB paBeH 100°.
5. Pa3HOCTb BYX O,l);HOCTOpOHHHX yrJIOB npH nepeCe'tleHHH
BYX nap8JIJIeJIbHbIX npHMbIX ceKym;eii paBHa 30°. HaH H-
Te 3TH yrJlbI.
6. B MEC LB - npHMoii. Qepe3 BepmHHY C rrpOBe)J;eHa npH-
MaR, KOTopaH nap8JIJIeJIbHa CTopOHe AB H o6paayeT C npH-
MOM AC yrOJI 36°. HaH)J;HTe pa3HOCTb LA H LC.
7. BepHo JIH YTBepm eHHe: qepe3 JII06bIe )J;Be TOqKH npOXO,l);HT
npHMaH, H npHTOM TOJlbKO o,l);Ha?
Bapuanm 2
1. TOqKa D - Cepe,l);HHa CTOpOHbI AC MEC. qepea TOqKH C H
D npOBe)J;eHbI npHMbIe n H m, napaJIJIeJIbHbIe npSlMoH AB.
IIepeceKalOTcH JIH npHMbIe n H m?
2. Qepe3 TO'tlKY, He JIemam;ylO Ha npHMOH b, npOBe eHbI Be
npHMbIe. CKOJIbKO H3 3THX rrpHMbIX nepeceKalOT npHMYID b?
3. CyMMa COOTBeTCTBeHHLIX yrJIOB rrpH nepeceqeHHH ABYX nap8JI-
JIeJIhHhIX npHMLIX ceK eH paBHa 140°. H Te 3TH yrJIhI.
4. HaH HTe Bce yrJIbI, o6pa30BaHHhIe rrpH nepeCeqeHHH ,lI;BYX
npHMbIX n H m ceK¥m;eH k, eCJIH O,l);HH H3 yr JIOB paBeH 50 0 .
5. Pa3HOCTb ,l);BYX O HOCTOpOHHHX yrJIOB npH nepeCeqeHHH
,l);BYX napaJIJIeJIbHbIX npHMbIX ceKym;eH paBHa 20°. HaH H-
Te 3TH yr JIbI.
6. B MEC LC - rrpHMoH. qepea BepmHHY B npOBe eHa npH-
MaSl, KOTopaH napaJIJIeJIbHa CTopOHe AC H o6paayeT C rrpH-
MOH AB yroJl 32°. HaH.n;HTe pa3HOCTb LA II LB.
7. BepHo JIH YTBep:m eHHe: Ha JII060M JIyqe OT ero Ha'tlaJla
MomHO OTJIOmHTb OTpeaOK, paBHbIH aHHOMY, H npHTOM
TOJlbKO O,lJ;HH?
CaMOCTOHTeJIbHaSl pa60Ta M 27
57
Bapuanm 3
1. TOQKa N - cepe HHa CTOpOHbI BC MBC. "lJepea TO KH B If
N npOBe eHbl npSlMble b H d, napaJIJIeJIbHble npHMoH AC.
IIepeceKalOTcSI JIH rrpHMble b H d?
2. qepea TO KY, He JIemam;ylO Ha npSlMOH n, npOBe,ZJ;eHbI TPH
npSlMbIe. CKOJIbKO H3 3THX npSlMhIX nepeceKalOT npHMYIO n?
3. CYMMa HaKpeCT JIe.iKa HX yrJIOB npH nepeCelIeHHH BYX
napaJIJIeJIbHbIX npHMblX ceKy eH paBHa 240°. Hati,ZJ;HTe 3TH
yr JIbl.
4. HaH HTe Bce yrJIbI, o6pa30BaHHble npH nepeCe"tleHHH BYX
npHMblX a H b ceKym;eti c, eCJIH O HH H3 yrJIOB paBeH 70°.
5. Pa3HocTb BYX O HOCTOpOHHHX yrJIOB npH nepeCe"tleHHH
BYX napaJIJIeJIhHhIX npHMbIX ceKym;e:n paBHa 50°. Ha:n H-
Te 3TH yr JIbl.
6. B MBC LB - rrpSlMoH. "lJepe3 BepmHHY C npOBe eHa rrpH-
MaSl, KOTopaSl napaJIJIeJIbHa CTopOHe AB H 06pa3yeT C rrpH-
MOM AC yrOJI 42°. HaH HTe pa3HOCTb LA H LC.
7. BepHo JIH YTBepmAeHHe: OT JII060ro JIyqa B 3a aHHYIO CTO-
POHY MomHO OTJIOmHTh yrOJI, paBHbIH aHHOMY Hepa3Bep-
HYTOMY yr JIY, H rrpHTOM TOJIbKO O HH?
Bapuanm 4
1. TOqKa D - cepe HHa CTOpOHbl BC MBC. qepea TOqKH C H D
rrpOBe eHLI rrpSlMble n H m, rrapaJIJIeJIbHbIe npHMoH AB. IIe-
peceKalOTCSI JIH npSlMLle n H m?
2. qepe3 TOqKY, He JIe.iKa ylO Ha npHMOH k, rrpOBe eHhI TPH
npHMhIe. CKOJIbKO H3 3THX rrpSlMbIX nepeceKalOT npHMYIO k?
3. CYMMa COOTBeTCTBeHHbIX yr JIOB rrpH rrepeCelIeHHH ABYX na-
paJIJIeJIbHhIX npHMbIX ceKy eii paBHa 160 0 . HaH HTe 3TH
yr JIbl .
4. HaHAHTe Bce yrJIbI, 06pa30BaHHbIe npH nepeCe'tIeHHH BYX
npHMbIX n H m ceKym;eii k, eCJIH O HH H3 yrJIOB paBeH 120°.
5. Pa3HOCTb ABYX OAHOCTOpOHHHX yrJIOB npH IIepece eHHH
BYX rrapaJIJIeJIbHbIX rrpSlMbIX ceKym;eH paRRa 40°. HaH H-
Te 8TH yrJIbI.
6. B ABC LC - npSlMoH. Qepe3 BepmHHY B npOBe eHa
npSlMaH, KOTopaH rrapaJIJIeJIbHa CTopOHe AC H 06paayeT c
rrpSlMoii AB yrOJI 47°. Hau,ZJ;HTe pa3HOCTb LA H LB.
7. BepHo JIH YTBepmAeHHe: aKCHOMbI - HCXO Hble nOJIome-
HHSI, Ha OCHOBe KOTOpbIX aJIee AOKa3blBalOTCH TeopeMbI?
58
CaMOCTOHTeJlbHaSl pa60Ta M 28
Bapuanm 1
1. ,n:aH MBC. CKOJIhKO npHMbIX, rrapaJIJIeJIhHbIX CTopOHe BC
M03tHO npOBeCTH qepea BepIIIHHy A?
2. HaH HTe Bce yrJIbI, 06paaoBaHHbIe rrpH nepeCeqeHHH ,n;BYX
rrpHMbIX n H m ceKym;eii k, eCJIH O,lJ;HH H3 yrJIOB Ha 90°
60JIhme ,n;pyroro.
3. LARC = 60°, a LBCD = 120°. MorYT JIH rrpHMbIe AB H CD
6hITb rrepeceKalO HMHcH?
4. ,n:Be rrapaJIJIeJIbHhIe npHMhIe nepeCeqeHbI ceKym;eH. BepHo
JIH YTBepm,n;eHHe: 6HcceKTpHcbI HaKpeCT Jle.iKanJ;HX yrJIOB
napaJlJIeJlbHbI?
5. BepHo JIH yrBep.iK,lJ;eHHe: qepe3 TOqKY, He JIemamylO Ha aH-
HOB npHMoii, npoxO HT Be rrpHMhIe, rrap8JIJIeJIbHLle 8.HHOH?
6. BHYTPH TyIToro LARC npOBe,n;eHbI napaJIJIeJIbHhIe JIy'tlH AD
H CE. LDAB = 132°; LBCE = 118°. HaH,lJ;HTe LARC.
7. lIa TOqeK A H B, JIema HX Ha O HOB H3 CTOpOH ,n;aHHoro
OCTporo yr JIa, rrpOBe,n;eHhI rreprreH HKY JIHpbI AC H BD. KO
BTOPOH CTopOHe yrJIa. LCAB = 125°. Haii,n;HTe LARD.
Bapuanm 2
1. aH tV3CD. CKOJIhKO npSlMhIX, napaJIJIeJIbHbIX CTopOHe BC,
MomHO npOBeCTH qepea BepmHHY D?
2. Haii HTe Bce yrJIhI, o6paaoBaHHhle rrpH nepeCeqeHHH ,n;BYX
rrpHMhIX a H b ceKy eH c eCJIH O HH 113 yrJIOB Ha 70° 60JIh-
me ,n;pyroro.
3. LNMP = 50°, a LMPS = 130°. MOryTJlH npHMhIeNMHPS
6bITL napaJIJIeJIbHbIMH?
4. ,n:Be rrapaJIJIeJlbHbIe npSlMhle nepeCeqeHhI ceKy eH. BepHo
JIH YTBepm,n;eHHe: 6HcceKTpHcbI HaKpeCT JIema HX yrJIOB
rrepeceKalOTcH?
5. BepHo JIH YTBep.iK,lJ;eHHe: eCJIH rrpHMaH nepeceKaeT O HY H3
BYX rrapaJIJIeJlbHhIX rrpHMbIX, TO oHa nepeceKaeT H pyrylO?
6. BHYTPH Tyrroro LARC npOBe,n;eHbI napaJIJIeJIbHbIe JIyqH AD
11 CE. LDAB = 126°; LBCE = 114°. HaH HTe LARC.
7. Ha TOqeK B H C, JIemam;HX Ha O,lJ;HOH H3 CTOpOH aHHoro
OCTporo yrJIa, rrpOBe,ZJ;eHbI nepneH HKYJIHpLI BD H CE KO
BTOPOH CTopOHe yrJIa. LBCE = 15°. HaH,lJ;HTe LCBD.
CaMOCTOHTeJIbHaa pa60Ta N 28
59
Bapuaum 3
1. aH C. CKOJIbKO rrpHMbIX, rrapaJIJIeJIhHbIX CTopOHe AB,
MomHO npOBeCTH "tIepe3 BepmHHY C?
2. HaH HTe Bce yrJIbI, o6paaoBaHHbIe rrpH nepece eHHH BYX
rrpHMbIX n Ii m ceKyrn;eH k, eCJIH O,l];HH 113 yrJIOB Ha 50° 60JIb-
me pyroro.
3. LABC = 70°, a LBCD = 110°. MorYT JIB npaMbIe AB H CD
6bITb rrepeceKalOW;HMHca?
4. Be napaJIJIeJIhHbIe rrpaMhIe rrepeCeqeHbI ceKYIIJ;eif. BepHo
JIH YTBepm eHHe: 6HcceKTpHcLI COOTBeTCTBeHHbIX yrJIOB
rrapaJIJIeJIbHbI?
5. BepHo JlH YTBepm eHHe: eCJIIf Be npaMbIe rrapaJIJIeJlbHbI
TpeTbeH rrpaMoii, TO OHH rrapaJIJIeJIbHbI?
6. BHYTPH Tynoro LABC npOBe.n;eHbI rrapaJIJIeJIbHhIe .rryqH AD
H CE. LDAB = 134°; LBCE = 106°. HaH)J;HTe LABC.
7. l1a TOqeK A H B, JIe.iKaW;HX Ha O.D;HOH H3 CTOpOH aHHoro
OCTporo yrJIa, rrpOBe eHbI nepneH HKYJlHPbI AC H BD KO
BTOpOH CTopOHe yrJIa. LCAB = 145°. HaH liTe LABD.
Bapuaum 4
1. AaH BCD. CKOJIbKO rrpHMbIX, nap8JIJIeJIbHbIX CTopOHe CD,
MomHO npOBeCTH Qepe3 BepllIHHY B?
2. Haii,l];HTe Bce yrJIbI, 06pa30BaHHbIe npH rrepeCeqeHHH BYX
rrpaMbIX a H b ceKy eH c, eCJIH O lfH H3 yrJIOB Ha 30° 60JIb-
me pyroro.
3. LNMP = 40°, a LMPS = 140°. MorYT JIH npHMLIe NMHPS
6bITL rrap8JIJIeJlbHbIMH?
4. ABe napaJIJIeJIhHbIe npaMbIe nepeCelIeHbI ceKy eii. BepHo
JIH YTBepm eHHe: 6HcceKTpHcbI COOTBeTCTBeHHbIX yrJIOB
nepeceKalOTcH?
5. BepHo JIH YTBepm.n;eHHe: lIepe3 TOqKY, He JIemam;yro Ha ,l];aH-
HOM npHMOH, npOXO BT TOJILKO o.n;Ha npHMaa, rrapaJIJIeJIb-
HaH aHHoii?
6. BHYTPH Tyrroro LABC rrpOBe eHbI naparrJIeJIbHbIe JIYlIH AD
H CE. LDAB = 128°; LBCE = 102°. Haii HTe LABC.
7. 113 TOlIeK B If C, JIe.iKa:rn;HX Ha o Hoii H3 CTOpOH .n;aHHoro
OCTporo yrJla, npOBe eHbI rreprreH HKYJIHpLI BD H CE KO
BTOpOH CTopOHe yrJIa. LBCE:= 75°. HaH)J;HTe LCBD.
60
CaMOCTOHTeJIbHaSi pa60Ta N2 29
BapuaJtm 1
1. BepHO JIB YTBepm)J;eHHe: eCJIH Be napaJIJIeJlbHbIe npHMhIe
nepeee'tleHbI ceKym;eH, TO HaKpeCT Jlema He yrJIbI paBHhI?
2. IIpSlMhle a II b nepneH HKYJIHpHbI K npHMoHp, npHMaH k ne-
peeeKaeT npHMYIO a. IIepeceKaeT JIH npHMaH k npHMYJO b?
3. Be nap8.JIJIeJlbHhIe npHMhIe nepeCeqeHhI ceKy eH. BepHo
JIH YTBepm eHlle: 611ceeKTpBchI O HOCTOpOHHHX yrJIOB rrep-
neH HKYJI.HpHbI?
4. Bre yr.1IhI, o6paooB8HHhIe llpH nepeooqemm: ,D;ByX npsIMbIX
m H n eeKym;eii p, ecJlB O,lJ;HH B3 yr JIOB B 5 paa 60JIbme APyroro.
5. IIpHMaH, npOXO)J;Hm;aH Qepe3 cepe HHY 6HceeKTpBcbI AD
MBC H neprreH HKYJIHpHa.H K AD, nepeceKaeT CTOPOHY AC
B TOtIKe M. BepHO JIB YTBepm eHlle: npHMbIe MD H AB ne-
peceKalOTcH?
6. B paBH06e)J;peHHoM TpeyrOJlbHIIKe C nepHMeTpOM 48 CM 60-
KOBaR CTopOHa OTHOCIITCH K OCHOBaHHIO KaK 5 : 2. HaH HTe
OCHOBaHHe TpeyrOJlbHHKa.
7. IIepHMeTp MBC paBeH 40 eM. Me)J;HaHa BD eJIHT MBC
Ha Ba TpeyrOJIhHHKa, nepHMeTphI KOTOpbIX paBHbI 28 CM B
24 CM. HaH HTe BD.
BapuaHm 2
1. BepHO JIB YTBepjK eHBe: eCJIIi rrpHMaH nepneH HKYJISIpHa K
O HOH H3 BYX napaJlJIeJIbHhIX npHMhIX, TO OHa nepneH H-
KYJlHpHa B K pyroH?
2. IIp.HMhIe a II b nepneH,Z\HKYJIHPHbI K npHMOH k, npHMaH P He ne-
peceKaeT npSlMYIO b. IIepeceKaeT JIB npHMaSi p npHMYIO a?
3. Be rrapaJIJIeJlhHbIe npSlMhIe nepeCe"CIeHbI ceKym;eii. BepHo
JIB YTBepm)J;eHHe: 6BcceKTpHChI COOTBeTCTBeHHhIX yrJIOB
napaJIJIeJIbHbI?
4. HaH BTe Bce yrJIbI, 06paaoBaHHble npH nepeCe"CIeHBH BYX
npHMhIX a H b ceKy eH c, eCJIB O)J;BH B3 yr JIOB B 4 pa3a 60JIb-
me pyroro.
5. IIpHMaH, npOXO Hm;a.H "CIepea cepe HHY 6HcceKTpllcbI BE
f1BCD H nepneH)J;IIKY JIHpHaSi K BE, nepeceKaeT CTOPOHY BD
B TOqKe F. BepHO JIB YTBepjK eHBe: npHMhIe BC II FE nepe-
ceKalOTcH?
6. B paBHo6e peHHoM TpeyroJlbHHKe C nepHMeTpOM 55 CM 60-
KOBaH CTopOHa OTHOCHTCSI K OCHOBaHHIO KaK 4 : 3. HaH HTe
60KOBYlO CTOPOHY TpeyrOJlhHBKa.
7. IIepBMeTp MBC paBeH 38 CM. Me HaHa AD eJIHT MBC
Ha ,l(Ba TpeyrOJILHHKa, nepHMeTpbI KOTOpbIX paBHhI 26 CM B
22 CM. HaH HTe AD.
CaMOCTOSlTeJIbHaH pa60Ta N2 29
61
BapuaJtm 3
1. BepHo JIH YTBep.m,a;eHHe: eCJIH .n;Be rrapaJIJIeJIbHbIe npHMbIe
nepeCeqeHbI ceKyw;eii, TO COOTBeTCTBeHHbIe yrJILI paBHbI?
2. IlpHMbIe a Ii b rreprreH)J;HKYJIHpHbI K IIpHMOii m, npHMaH n rre-
peceKaeT rrpHMyIO b. IIepeceKaeT JIH np.HMaSI n rrpHMyro a?
3. Be napaJIJIeJIbHbIe npHMbIe nepeCetleHbI ceKyw;eii. BepHo
JIH YTBepjK eHHe: 6HcceKTpHcbI O.n;HOCTOpOHHHX yrJIOB rra-
paJIJIeJIbHhI ?
4. HaH,D;HTe Bce yrJIbI, 06paaoBaHHbIe npH rrepeCeqeHHH BYX
npSIMbIX m H n ceKyw;eif p, eCJIH O,n;HH Ha yr JIOB B 3 paaa MeHb-
me ,n;pyroro.
5. IIpHMaH, npOXO Hw;aH qepea cepe HHY 6HcceKTpHcbI AD
AABC H nepneH.n;HKYJIHpHaSl K AD, rrepeceKaeT CTOPOHY AC
B TOtlKe M. BepHO JIM yTBepmAeHHe: rrpHMbIe MD H AB rra-
paJIJIeJIbHbI?
6. B paBH06e,n;peHHoM TpeyroJIhHHKe C nepHM.eTpOM 56 CM 60-
KOBaH CTopOHa OTHOCHTCH K OCHOBaHHIO KaK 5 : 4. Hati,l1;HTe
OCHOBaHHe TpeyrOJIbHHKa.
7. IIepHMeTp AABC paBeH 39 CM. Me,ll;HaHa BD .n;eJIHT AABC Ha
,n;Ba TpeyrOJIbHHKa, rrepHMeTpbI KOTOpbIX paBHbI 26 CM H
23 CM. Haif,lJ;HTe BD.
Bapuanm 4
1. BepHo JIH YTBep)K,n;eHHe: eCJIH ,n;Be napaJIJIeJIbHbIe rrpHMbIe
IIepeCetleHbI ceKym;eii, TO CYMMa O,D;HOCTOpOHHHX yrJIOB paB-
Ha 180°?
2. IIpHMbIe a H b rreprre HKYJIapHbI K IIpHMoif n, IIpHMaH p He ne-
peceKaeT IIPHMYIO a. IlepeceKaeT JIll rrpHMaH p rrpSlMYIO b?
3. Be IIapaJIJIeJIbHbIe rrpHMbIe rrepeCeqeHbI ceKyw;eH. BepHo
JIH YTBepm,ll;eHHe: 6HcceKTpHchI HaKpeCT JIemaW;HX yrJIOB
rrap8.JIJIeJIbHbI?
4. HaH,lI;HTe Bce yrJIbI, o6paaoBaHHbIe npH rrepeCeqeHHH BYX
rrpHMbIX a H b ceKyw;eH c, eCJIH O,n;HH H3 yrJIOB B 2 pa3a MeHb-
me ,n;pyroro.
5. IIpHMaH, IIpOXO Hm;aH tlepe3 cepe,n;HHY 6HcceKTpHcbI BE
!!iBCD H rreprreHtJ;HKYJIHpHaH K BE, rrepeceKaeT CTOPOHY BD
B TOqKe F. BepHo JIH YTBepjK eHHe: rrpHMbIe EF H BC napaJI-
JIeJIbHbI?
6. B paBHo6e,n;peHHoM TpeyrOJIbHHKe C rrepllMeTpoM 65 CM 60KO-
BaH CTopOHa OTHOCHTCSI K OCHOBaHHIO KaK 5 : 3. HaH HTe 60KO-
Byro CTOpoHY TpeyrOJIbHHKa.
7. IIepHMeTp AABC paBeH 42 CM. Me,n;HaHaAD eJIHT BC Ha
.n;Ba TpeyrOJIbHHKa, nepHMeTpbI KOTOphIX paBHbI 28 CM II
22 CM. Haii,lJ;HTe AD.
62
CaMOCTOHTeJIbHaH pa60Ta N2 30
Bapuanm 1
1. HaH)J;HTe LC C, eCJIH LA = 40°, a LB B 2 paaa 60JIbme.
2. HaH HTe MeHbmHH yrOJI TpeyrOJIbHHKa, eCJIH H3BeCTHO, 'tITO
OTHoweHHe rp CHbIX Mep yrJIOB: 2 : 3 : 4.
3. BepHo JIH YTBep.iK eHHe: BHemHHH yrOJI TpeyrOJILHHKa pa-
BeH CYMMe BYX yr JIOB TpeyrOJIbHHKa, He CMeJKHbIX C HHM?
4. BepHO JIB YTBepjK eHHe: yrJIbI npH OCHOBaHHH paBHo6e -
peHHoro TpeyrOJIbHHKa TynbIe?
5. BHcceKTpHcLI LA H LB C nepeceKalOTCH B To'tJ:Ke M.
HaH HTe LAMB, eCJIH LA = 56°, a LB = 98°.
6. O HH H3 BHemHHX yrJIOB paBHo6e peHHoro TpeyrOJIbHHKa
paBeH 116°. HaH HTe yrOJI npH OCHOBaHHH p8BHo6e peHHo-
ro TpeyrOJIbHHKa.
7. B TpeyroJlbHHKe npOBe eHa Me HaHa JIHHOH 8 CM. Me Ha-
Ha eJIHT aHHhIR TpeyrOJIbHHK Ha ,lI;Ba TpeyrOJIbHHKa C ne-
pHMeTpaMH 25 CM H 27 CM. HaH HTe nepHMeTp aHHoro
TpeyrOJILHHKa.
Bapuanm 2
1. HaH HTe LA C, eCJIH LB = 120°, a LC B 3 paaa MeHbme.
2. Ha HTe 60JILmHH yrOJI TpeyrOJILHHKa, eCJIH H3BeCTHO, 'tJ:TO
OTHomeHHe rpap;ycHhIH Mep yr JIOB TpeyrOJIbHHKa: 4 : 3 : 2.
3. BepHo JIH YTBepjK eHHe: Kam bIH yrOJI paBHOCTopOHHero
TpeyrOJILHHKa paBeH 50°?
4. BepHO JIH YTBepjK eHHe: BHemHHH yroJI TpeyrOJIbHHKa - yroJI,
CMe.iKHbIH C KaKHM-HH6Y b yrJIOM 3Toro TpeyrOJIbHHKa?
5. BHcceKTpHchI LC H LD !!iBCD nepeceKalOTCH B TO Ke N.
HaH,lI;HTe LCND, eCJlH LC = 52°, a LD = 106°.
6. O HH H3 BHemHHX yrJIOB p8BHo6e peHHoro TpeyrOJIbHHKa
paBeH 112°. HaH HTe yrOJI, IIpOTHBOJIejKa HH OCHOBaHH1O.
7. B TpeyroJlbHHKe npOBe eHa Me,n;HaHa JIHHOH 7 CM. Me,n;Ha-
Ha ,lI;eJIHT aHHbIH TpeyrOJIbHHK Ha ,lJ;Ba TpeyrOJIbHHKa C ne-
pHMeTpaMH 23 CM H 25 CM. Haii HTe nepHMeTp ,lI;aHHoro
TpeyrOJIhHHKa.
CaMOCTOHTeJIbHaH pa60Ta NQ 30
63
Bapuaum 3
1. Haft.n;HTe LC MBC, eCJIH LA = 30°, a LB B 2 pa3a 60JIbme.
2. HaM.n;HTe MeHbmHH yrOJI TpeyrOJIhHHKa, ecJIH H3BeCTHO, qTO
OTHomeHHe rp8)JyCHbIX Mep yrJlOB: 1 : 3 : 5.
3. BepHo JIH YTBep.iK,n;eHHe: Kam hIH yrOJI paBHOCTopOHHero
TpeyrOJIbHHKa paBeH 60°?
4. BepHo JIll YTBep:>K eHHe: CYMMa yrJIOB TpeyrOJIbHHKa paBHa
200°1
5. BHcceKTpHcbI LA H LB MBC nepeceKalOTCSI B TOqKe M.
HaI:i,n;HTe LAMB, eCJIH LA = 48°, a LB = 104°.
6. O.n;HH H3 BHemHHX yr JIOB paBHo6e,n;peHHoro TpeyrOJIbHHKa
paBeH 118". Haii.D;HTe yrOlt npH OCHOBaHHH paBHo6e,n;peHHo-
ro TpeyrOJIbHHKa.
7. B TpeyrOJIbHHKe npOBe,n;eHa Me,n;HaHa JIHHOH 6 CM. Me,n;Ha-
Ha ,n;eJIHT ,n;aHHbIH TpeyrOJIbHHK Ha ,n;Ba TpeyrOJIbHHKa c ne-
pHMeTpaMH 24 CM H 28 CM. Haif,n;HTe nepHMeTp aHHoro
TpeyrOJIbHHKa.
Bapuaum 4
1. Haif,n;HTe LA ABC, eCJIH LB = 90°, a LC B 3 pa3a MeHbIIIe.
2. Haii,n;HTe 60JIbIIIHH yrOJI TpeyrOJIbHHKa, eCJIH H3BeCTHO, 'tITO
OTHOIIIeHHe rpa,n;ycHbIH Mep yr JIOB TpeyrOJIbHHKa: 3 : 5 : 1.
3. BepHo JIH YTBepm,n;eHHe: BHemHHH yrOJI TpeyrOJIbHHKa pa-
BeH pa3HOCTH iJ:BYX yrJIOB TpeyrOJIbHHKa, He CMemHbIX C
HHM?
4. BepHo JIH YTBepm,n;eHHe: yrJIbI npH OCHOBaHHH paBHo6e,n;-
peHHoro TpeyrOJIbHHKa oCTpbIe?
5. BHcceKTpHcbI LC H LD BCD nepeceKalOTCH B TO'tlKe N.
HaH,n;HTe LCND, eCJIH LC = 44°, a LD = 112°.
6. O.n;HH H3 BHemHHX yrJIOB paBHo6e,n;peHHoro TpeyrOJIbHHKa
paBeH 114 0 . HaH,n;HTe yrOJI, rrpOTHBOJIema:rn;HH OCHOBaHHIO.
7. B TpeyrOJIbHHKe npOBe,n;eHa Me HaHa ,n;JIHHOH 11 CM. Me-
,n;HaHa ,n;eJIHT ,n;aHHbIH TpeyrOJIbHHK Ha ,n;Ba TpeyrOJIbHHKa C
rrepHMeTpaMH 26 CM H 28 CM. HaH HTe rrepHMeTp aHHoro
TpeyrOJIbHHKa.
64
CaMOCTOHTeJIbHaH pa60Ta N2 31
Bapuanm 1
1. HaH HTe yrOJI rrpH OCHOBaHHH paBHo6e,n;peHHoro TpeyrOJIh-
HHKa, eCJIH 3TOT yrOJi B 2 paaa 60JIhllIe yrJIa, rrpOTHBOJIema-
w;ero OCHOBaHHIO.
2. HaH,n;HTe yrJIbI paBHo6e,n;peHHoro TpeyrOJIhHHKa, eCJIH
O,n;HH H3 ero yrJIOB paBeH 92°.
3. B paBHo6e,n;peHHoM ABC C OCHOBaHHeM AC npOBe,n;eHa
6HcceKTpHca AD. HaH,n;HTe LADC, eCJIH LC = 48°.
4. Me,n;HaHa AM MBC paBHa TIOJIOBHHe CTOpoHbI BC. BepHo JIH
YTBepjK eHHe: MBC - TYrroyrOJIbHbIH?
5. BepHo JIH YTBepmtJ;eHHe: 6HcceKTpHca BHemHero yrJIa rrpH
BepmHHe paBHo6e,n;peHHoro TpeyrOJIbHHKa, rrpOTHBOJIema-
m;eH OCHOBaHHIO, napaJIJIeJIbHa OCHOBaHHIO?
6. Ha 60KOBbIX CTopOHax paBHo6e,n;peHHoro MBC C OCHOBa-
HHeM AC OTJIO.iKeHbI paBHbIe OTpe3KH AM H CN. BD - Me-
,n;HaHa MBC - rrepeceKaeT OTpe30K MN B TO'tIKe O. BepHO
JIH YTBepm,n;eHHe: BO - MetJ;HaHa MBN?
7. B paBHo6e peHHoM TpeyrOJIbHHKe C rrepHMeTpOM 48 CM oc-
HOBaHHe OTHOCHTCH K 60KOBOH CTopOHe KaK 2 : 3. HaH HTe
,n;JII-IHY OCHOBaHHH.
BapuaHm 2
1. Haii,n;HTe yrOJI rrpH OCHOBaHHH paBH06e,n;peHHoro TpeyrOJIb-
HHKa, eCJIH STOT yrOJI B 3 pa3a MeHbme yrJIa, npOTHBOJIema-
m;ero OCHOBaHHIO.
2. HaH,n;HTe yrJIbI paBHo6e,n;peHHoro TpeyrOJIbHHKa, eCJIH
O HH H3 ero yrJIOB paBeH 98°.
3. B paBHo6e,n;peHHoM BCD C OCHOBaHHeM BD rrpOBe,n;eHa
6HcceKTpHca DE. HaH HTe LBED, eCJIH LB = 52°.
4. Me HaHa BN MBC paBHa IIOJIOBHHe CTOpOHbI AC. BepHo JIH
yrBepmtJ;eHHe: AABC - oCTpoyrOJIbHbIH?
5. BepHo JIll YTBepm,l];eHHe: B oCTpoyrOJIbHOM TpeyrOJIbHHKe
Bce TPH yrJIa oCTpbIe?
6. Ha 60KOBbIX CTopOHax paBHo6e,n;peHHoro MBC C OCHOBa-
HHeM AC OTJIOmeHbI paBHLIe OTpe3KH AS H CT. BD - 6HC-
ceKTpHca MBC - rrepeceKaeT OTpe30K S T B TOqKe P. BepHo
JIH yrBepm,n;eHHe: BP - Me,l];HaHa MBN?
7. B paBHo6e peHHoM TpeyrOJIbHHKe c nepHMeTpOM 44 CM oc-
HOBaHHe OTHOCHTCH K 60KOBOH CTopOHe KaK 3 : 4. HaH HTe
,n;JIHHY 60KOBOH CTOpOHbI.
CaMOCTO.fITeJIbHaH pa60Ta N2 31
65
Bapuanm 3
1. HaH,lJ;HTe yrOJI rrpH OCHOBaHHH paBHo6e,n;peHHoro TpeyrOJIb-
HHKa, eCJIH 3TOT yrOJI B 4 pa3a 60JIbme yrJIa, rrpOTHBOJIema-
m;ero OCHOBaHHIO.
2. HaHAHTe yrJIhI paBHo6e,ZJ;peHHoro TpeyrOJIbHHKa, eCJIH
O,ZJ;HH M3 ero yrJIOB paBeH 88°.
3. B paBH06e,lJ;peHHOM ABC C OCHOBaHHeM AC npOBe,n;eHa
6HcceKTpHca AD. HaH,lJ;HTe LADC, eCJIH LC = 54°.
4. Me,n;lIaHa AM MEC paBHa IIOJIOBHHe CTOpOHbI BC. BepHo JIll
YTBepm eHHe: MEC - oCTpoyrOJIhHbIii?
5. BepHo JIH YTBepm,lJ;eHHe: B npHMoyrOJIbHOM TpeyrOJIbHHKe
,lJ;Ba npHMhIX yr JIa ?
6. Ha 60KOBbIX CTopOHax paBH06e,ZJ;peHHoro MEC C OCHOBa-
HHeM AC OTJIOmeHbI paBHbIe OTpeaKH AM If CN. BD - Me-
,lJ;HaHa c - nepeceKaeT OTpe30K MN B TOqKe O. BepHO
JIM YTBepm,lJ;eHHe: BO - BbICOTa MBN?
7. B paBH06e,lJ;peHHOM TpeyrOJIbHHKe C nepHMeTpOM 60 CM oc-
HOBaHHe OTHOCHTCH K 60KOBOH CTopOHe KaK 2 : 5. Haii,ZJ;HTe
,lJ;JIHHY OCHOBaHH.fI.
Bapuanm 4
1. HaH,lJ;HTe yrOJI rrpH OCHOBaHHH paBHo6e,n;peHHoro TpeyrOJIb-
HHKa, eCJIH 3TOT yrOJi B 7 paa MeHbme yrJIa, npOTHBOJIema-
m;ero OCHOBaHHIO.
2. HaHAHTe yrJIhI paBHo6e,ZJ;peHHoro TpeyrOJIbHHKa, eCJIH
O,n;HH H3 ero yr JIOB paBeH 96°.
3. B paBH06e,lJ;peHHOM BCD C OCHOBaHHeM BD rrpOBeAeHa
6HcceKTpHca DE. HaH,ZJ;MTe LBED, eCJIH LB = 46°.
4. Me,lJ;HaHa BN MEC paBHa IIOJIOBHHe CTOpoHbI AC. BepHO JIM
YTBepm,lJ;eHHe: MEC - TynOyrOJIbHhIH?
5. BepHo JIM YTBepm,n;eHHe: B TyrroyrOJIbHOM TpeyrOJIbHHKe
Ba TYIIbIX yr JIa ?
6. Ha 60KOBhIX CTopOHax paBHo6e,n;peHHoro C C OCHOBa-
HHeM AC OTJIOmeHbI paBHhIe OTpe3KH AS H CT. BD - 6HC-
ceKTpHca MEC - rrepeceKaeT OTpeaOK S T B TOqKe P. BepHO
JIM YTBepm,n;eHHe: BP - BbICOTa MBN?
7. B paBH06e,lJ;peHHOM TpeyrOJIbHHKe C nepHMeTpOM 52 CM oc-
HOBaHMe OTHOCHTCH K 60KOBOH CTopOHe KaK 3 : 5. HaH,ZJ;HTe
,n;JIHHY 60KOBOii CTOpOHbI.
66
CaMOCTOSlTeJIhHaH pa60Ta N 32
BapuaKm 1
1. B paBH06e,l(peHHOM MBC C OCHOBaHHeM AC rrpOBe eHa
6HcceKTpHca AD. HaH HTe LABC, eCJIH LADB = 120°.
2. HaH,ZJ;IITe yrJIbI paBH06e peHHoro TpeyrOJIbHIIKa, eCJIH yrOJI
npH OCHOBaHIIH B 3 paaa MeHhIIIe BHemHero yr JIa, CMemHO-
ro C HIIM.
3. CpaBHHTe yrJIhI AABC H BhIHCHIITe, MomeT JIH 6hITb LA TY-
TIhIM, eCJIH AB > BC > AC?
4. BepHo JIll YTBepm eHHe: eCJIH 6HcceRTpllca BHemHero yrJIa
TpeyrOJIhHHKa rrapaJIJIeJIhHa CTopOHe TpeyrOJIbHHKa, TO
TpeyrOJIbHHK paBHo6e,n;peHHhIH?
5. B paBH06e peHHoM AABC C OCHOBaHlleM AC 6HcceKTpHchl
LA H LC rrepeceKaIOTCH B TOtIKe o. BepHo JIH YTBepm eHlle:
MOC - paBHocTopOHHHH?
6. IIepHMeTp paBHo6e peHHoro TpeyrOJIbHHKa B 4 paaa 60JIh-
me OCHOBaHHH II Ha 10 CM 60JIbme 60KOBOH CTOpoHhl. HaH-
HTe TIepllMeTp TpeyrOJIhHHKa.
7. BepHo JIH YTBepm eHlle: cepe HHhI CTOpOH paBHo6e peHHo-
ro TpeyrOJIhHHKa HBJIHIOTCH BepmHHaMH pyroro paBHO-
6e peHHoro TpeyrOJIhHHKa?
Bapuanm 2
1. B paBH06e,n;peHHoM I1BCD C OCHOBaHHeM BD npoBe eHa
6HcceKTpHca BE. HaH HTe LBCD, eCJIH LBEC = 105°.
2. HaH HTe yr JIhl paBHo6e,ZJ;peHHoro TpeyrOJIhHHKa, eCJIII yrOJI
rrpll OCHOBaHHH B 4 paaa MeHhme BHemHero yr JIa, CMemHO-
ro C HHM.
3. CpaBHIITe yrJIhI AABC H BhISlCHIITe, MomeT JIll 6bITb LB
OCTphIM, eCJIH AB > BC > AC?
4. BepHo JIll YTBepm eHHe: eCJIH 6HcceKTpHca BHemHero yrJIa
TpeyrOJIbHHKa napaJIJIeJIhHa CTopOHe TpeyrOJIhHHKa, TO
TpeyrOJIbHHK paBHocTopOHHHH?
5. B paBH06e peHHoM AABC C OCHOBaHHeM AC 6HcceKTpHchI
LA H LC rrepeceK810TCH B TO"tlKe o. BepHo JIll YTBepm,n;eHHe:
MOC - paBH06e peHHbIH?
6. IIepHMeTp paBHo6e,n;peHHoro TpeyrOJIbHHKa B 5 pa3 60JIhIIIe
OCHOBaHHH H Ha 12 CM 60JIhme 60KOBOH CTOpOHLI. HaH IITe
nepHMeTp TpeyrOJIbHHKa.
7. BepHo JIH YTBepjK eHHe: B TpeyrOJIhHHKe npOTHB 60JIbmeH
CTOpOHLI JIemHT 60JIhmHH yroJI?
CaMOCTOHTeJIbHaH pa60Ta N2 32
67
Bapuanm 3
1. B paBHo6e,ZJ;peHHoM ABC C OCHOBaHHeM AC npOBe eHa
6HcceKTpHca AD. Haif,ZJ;HTe LABC, eCJIH LADC = 60°.
2. HaH HTe yr JIbI paBHo6e,ZJ;peHHoro TpeyrOJIbHHKa, eCJIH yrOJI
npH OCHOBaHHH B 5 pa3 MeHbme BHeIllHero yrJIa, CMemHoro
C HHM.
3. CpaBHHTe yrJIbI ABC H BbIHCHHTe, MomeT JIH 6bITb LA
npHMbIM, eCJIH AB > BC > AC?
4. BepHo JIH YTBepm)J;eHHe: eCJIH 6HcceKTpMca BHeIllHero yrJIa
TpeyrOJIbHHKa napaJIJIeJIhHa CTopOHe TpeyrOJIbHHKa, TO
TpeyrOJIbHHK paBHo6e.n;peHHbIH?
5. B paBHo6e.n;peHHoM MBC C OCHOBaHlfeM AC 6lfCCeKTpHcbI
LA H LC nepeceKalOTCH B TO'tlKe O. BepHo JIH YTBepm.n;eHHe:
MOC - nps.rMoyrOJIbHbIH?
6. IIepHMeTp paBHo6e.n;peHHoro TpeyrOJIbHHKa B 7 pa3 60JIbme
OCHOBaHlfH H Ha 12 CM 60JIbllIe 60KOBOH CTOpOHbI. HaH HTe
nepMMeTp TpeyrOJIhHHKa.
7. BepHo JIH yrBepm.n;eHHe: B npHMoyrOJIbHOM TpeyrOJIbHHKe
rHnOTeHY3a 60JIbllIe KaTeTa?
Bapuanm 4
1. B paBHo6e.n;peHHoM BCD C OCHOBaHHeM BD npOBe.n;eHa
6HcceKTpHca BE. Haii.n;HTe LBCD, eCJIH LBED = 75°.
2. Haif,ZJ;HTe yrJIbI paBHo6e.n;peHHoro TpeyrOJIbHHKa, eCJIH yrOJI
nplf OCHOBaHHH B 8 pa3 MeHbme BHemHero yr JIa, CMemHoro
C HHM.
3. CpaBHHTe yrJIbI ABC H BbIHCHHTe, MomeT JIH 6bITb LC
OCTpbIM, eCJIH AB > BC > AC?
4. BepHo JIM YTBepm,n;eHHe: eCJIH 6HcceKTpHca BHeIllHero yrJIa
TpeyrOJIbHHKa napaJIJIeJIbHa CTopOHe TpeyrOJIbHHKa, TO
TpeyrOJIbHHK pa3HocTopOHHHH?
5. B paBHo6e,n;peHHoM MBC C OCHOBaHHeM AC 6HcceKTpMchI
LA If LC nepeceKalOTCH B TO'tlKe o. BepHO JIB YTBepm,n;eHHe:
MOC - TYnoyrOJIbHbIH?
6. IIepHMeTp paBHo6e peHHoro TpeyrOJIbHHKa B 9 paa 60JIbllIe
OOHOBaHHH If Ha 10 CM 60JIbme 6oKoBoif CTOpOHbI. Hal1,lJ;HTe
nepHMeTp TpeyrOJIbHHKa.
7. BepHo JIll yrBepm,lJ;eHHe: B TpeyrOJIbHlfKe npOTHB 60JIbllIerO
yrJIa JIemHT 60JIhllIaH cTopoHa?
68
CaMOCTOHTeJIbHaH pa60Ta N2 33
BapuaHm 1
1. HaaoBHTe 6oJIbmylO CTOPOHY MEC, eCJIH LA 60JIbllIe, qeM
LB, a LB paBeH LC.
2. Cym;ecTByeT JIH TpeyrOJIbHHK co CTopOHaMH 2 CM, 3 CM H
5 CM?
3. B paBHo6e,n;peHHoM TpeyrOJIbHHKe o,n;Ha CTopOHa paBHa
15 CM, a pyraH 5 CM. qeMY paBHa )J;JIHHa oCHoBaHHH?
4. HaH HTe CTOPOHY paBHo6e peHHoro TpeyrOJIbHHKa, eCJIH
)];Be ,n;pyrHe CTOpoHbI paBHhI 12 CM H 6 CM.
5. ABa BHeIDHHX yr JIa TpeyrOJIbHHKa npH pa3HbIX BepmHHax
paBHbI. IIepHMeTp TpeyroJlbHHKa paBeH 72 CM, a O Ha H3
CTOpOH paBHa 14 CM. Hai1)J;HTe ,lI;Be ,n;pyrHe CTOpOHhI Tpe-
yrOJIbHHKa.
6. BepHo Jill YTBepm eHHe: Kam,n;aH CTopOHa TpeyrOJIbHHKa
MeHbIlle CYMMbI BYX ,n;pyrHx CTOpoH?
7. CYMMa )];BYX yr JIOB TpeyrOJIbHHKa paBHa 80°, a O,n;HH 113
3THX yrJIOB 60JIhme pyroro Ha 60°. HaH)];HTe MeHbIDHH
yrOJI TpeyrOJIhHHKa.
BapU4Hm 2
1. HaaoBHTe MeHhWYIO CTOPOHY CD, eCJIH LB MeHhIIIe, eM
LC, a LC paBeH LD.
2. Cym;ecTByeT JIH TpeyrOJIhHHK co CTopOHaMH 3 CM, 3 CM H
4 CM?
3. B paBHo6e peHHoM TpeyrOJIbHHKe O Ha CTopOHa paBHa
12 CM, a pyra.H 4 CM. qeMY paBHa ,n;JlHHa 6oKoBOH CTOpoHbI?
4. Haii,n;HTe CTOPOHY paBH06e)];peHHOro TpeyrOJIhHHKa, eCJIH
Be pyrHe CTOpOHhI paBHhI 10 CM H 5 CM
5. ,l];Ba BHeWHHX yrJIa TpeyrOJIhHHKa npH paaHhIX BepmHHax
paBHbI. ITepHMeTp TpeyrOJIhHHKa paBeH 64 CM, a O,ZJ;Ha H3
CTOpOH paBHa 16 CM. Haif,ZJ;HTe ,l];Be ,n;pyrHe CTOpOHbI Tpe-
yrOJIbHHKa.
6. BepHo JIM YTBepJK,lI;eHHe: Kam a.H CTopOHa TpeyrOJIhHHKa
MeHbme pa3HOCTH ,l];BYX PyrHx CTOpOH?
7. CYMMa BYX yrJIOB TpeyrOJILHHKa paBHa 100°, a O,l];IIH 113
3THX yrJIOB MeHhme pyroro Ha 40°. HaH HTe 60JIhillHH
yrOJI TpeyrOJIhHHKa.
CaMOCTOSITeJIbHaH pa60Ta N2 33
69
Bapuanm 3
1. HaaOBHTe 60JIbillYlO CTOPOHY !1MNP, eCJIH LM 60JIbIIle,
qeM LN, a LN paBeH LP.
2. Cy ecTByeT JIH TpeyrOJIbHHK co CTopOHaMH 3 CM, 4 CM H
7 CM?
3. B paBHo6e)J;peHHoM TpeyrOJIbHHKe O)J;Ha CTopOHa paBHa
13 CM, a pyraSI 3 CM. qeMY paBHa )J;JIHHa oCHOBaHH.H?
4. HaH,lJ;HTe CTOPOHY paBHo6e peHHoro TpeyrOJIbHHKa, eCJIH
)J;Be ,lJ;pyrHe CTOpOHhI paBHbI 14 CM H 7 CM.
5. ,lJ;Ba BHemHHX yrJIa TpeyrOJIbHHKa npH pa3HbIX BepmHHax
paBHbI. IIepHMeTp TpeyrOJIbHHKa paBeH 75 CM, a O,lJ;Ha H3
CTOpOH paBHa 13 CM. HaH,lI;HTe )J;Be )J;pyrHe CTOpOHbI Tpe-
yrOJIbHHK&.
6. BepHo JIH YTBepm)J;eHHe: IIpOTHB 60JIbmeH CTOpOHbI JIemHT
60JIblliHH yroJI?
7. CYMMa ,lI;BYX yr JIOB TpeyrOJIbHHKa paBHa 110°, a O,lJ;HH H3
3THX yrJIOB 60JIbIIle )J;pyroro Ha 50°. Haft,lJ;HTe MeHbmHH
yrOJI TpeyrOJIbHHKa.
BapuaHm 4
1. HaaoBHTe MeHbillYIO CTOPOHY CDE, eCJIH LC MeHbille, qeM
LD, a LD paBeH LE.
2. Cym;ecTByeT JIH TpeyrOJIbHHK co CTopOHaMH 4 CM, 5 CM II
6 CM?
3. B paBHo6e,D;peHHOM TpeyrOJIbHHKe O)J;Ha CTopOHa paBHa 5 CM,
a ,D;pyraH 10 CM. 'tJeMY paBHa )J;JIHHa 60KOBOH CTOpOHbI?
4. Haii,D;HTe CTOPOHY paBH06e,lJ;peHHOro TpeyrOJIbHHKa, eCJIH
,lJ;Be ,lJ;pyrHe CTOpOHbI paBHbI 16 CM H 8 CM.
5. ABa BHemHHX yrJIa TpeyrOJIbHHKa IIPH pa3HbIX BepllIHHax
paBHbI. IIepHMeTp TpeyrOJIbHHKa paBeH 73 CM, a OtJ;Ha 113
CTOpOH paBHa 17 CM. HaH,lJ;HTe ,lJ;Be ,lJ;pyrHe CTOpOHbI Tpe-
yrOJIbHHKa.
6. BepHo JIH YTBepm)];eHHe: B rrpHMoyrOJIbHOM TpeyrOJIbHHKe
KaTeT 60JIbme rHIIoTeHY3hI?
7. CYMMa ,D;BYX yrJloB TpeyroJlbHHKa paBHa 120°, a O,D;HH H3
3THX yr JIOB MeHbllIe )J;pyroro Ha 20°. Hai1:,lI;HTe 60JIbillllH
yrOJI TpeyrOJIbHHKa.
70
CaMOCTOSlTeJIhHaH pa60Ta N2 34
Bapuanm 1
1. B paBH06e)\peHHOM BCD C OCHOBaHHeM BD npOBe eHa BhI-
COTa BE. HaH HTe LDBE, eCJIH LC = 52°.
2. O)\HH H3 yrJIOB npHMoyrOJIhHOrO TpeyrOJIhHHKa paBeH 60°, a
CYMMa rHrrOTeHY3hI H MeHhmero H3 KaTeTOB paBHa 31,5 CM.
HaH,lI;HTe rHnOTeHY3Y TpeyrOJIhHHKa.
3. B npHMoyrOJIhHOM MBC c rrpSiMbIM LC BHemHHH yrOJI npH
BepIIlHHe A paBeH 120°, AB - AC = 8 CM. HaH HTe AB.
4. 113 cepe HHhI D CTOpOHhI BC paBHOCTopoHHero MBC rrpo-
Be eH nepneHAHKYJIHp DM K npHMoH AC. HaH HTe AM,
eCJIH AB = 18 CM.
5. BepHo JIH YTBepm,ll;eHHe: B paBHo6e peHHoM TpeyrOJIhHHKe
BhICOThI, npOBe eHHhIe 113 BeplliHH OCHOBaHHH, paBHhI?
6. CYMMa KaTeTOB TpeyrOJIhHHKa paBHa 7 CM, a O HH KaTeT
60JIhme pyroro Ha 1 CM. HaH HTe nJIOID;Mh TpeyrOJIbHHKa.
7. ,n:a.H OCTpoyrOJIbHhIH MBC, BD - BbICOTa TpeyroJIhHHKa. Ha
CKOJIhRO qaCTeH eJIHT llJIOCROCTb npHMbIe AB, BC, CA H BD?
Bapuanm 2
1. B paBHo6eApeHHoM MBC C OCHOBaHHeM AC rrpOBe,ll;eHa BbI-
COTa AD. HaH HTe LCAD, eCJIH LB = 56°.
2. O,ll;HH H3 yr JIOB npHMoyrOJIhHOrO TpeyrOJIbHHKa paBeH 60°, a
CYMMa rHrrOTeHY3bI H MeHhmero H3 KaTeTOB paBHa 37,5 CM.
HaH HTe MeHblliHH H3 RaTeTOB.
3. B npHMoyrOJIhHOM MBC C rrpSlMhIM LB BHemHHH yrOJI npH
BepmHHe C paBeH 120°, AC - BC = 6 CM. HaH HTe AC.
4. 113 cepe HHhI S C OpOHhI NP paBHOCTopOHHero MNP rrpo-
Be eH rreprreH,lI;HKYJIHp S T K rrpHMOH MP. Ha HTe MT, ec-
JIH MN = 20 CM.
5. BepHo JIH YTBep)K eHHe: B paBH06e,ll;peHHoM TpeyrOJIhHHKe
Me HaHbI, npoBe eHHhIe H3 BepmllH OCHOBaHHH, paBHhI?
6. CYMMa KaTeTOB TpeyrOJIhH:HKa paBHa 14 CM, a O)\HH KaTeT
MeHhme .n;pyroro Ha 2 CM. Haii,ll;HTe llJIOID;Mh TpeyrOJIbHHKa.
7. .D:aH TynoyrOJIbHhIH I1BCD, CE - 6HcceKTpHca TpeyrOJIbHH-
Ra. Ha CKOJIhKO qaCTeH )\eJIHT nJIOCROCTb npHMbIe BC, CD,
BD H CE?
CaMOCTOHTeJlbHa.s pa60Ta N2 34
71
Bapuanm 3
1. B paBHo6e peHHoM I1.BCD C OCHOBaHHeM BD rrpOBe eHa BbI-
COTa BE. HaH)];HTe LDBE, eCJIH LC = 58°.
2. O HH H3 yrJIOB npHMoyrOJIbHOrO TpeyrOJIbHHKa paBeH 60 0 , a
CYMMa rHnOTeHY3bI Ii MeHbmero 1-13 KaTeTOB paBHa 34,5 CM.
HaH HTe rHIIOTeHY3Y TpeyrOJIbHHKa.
3. B npHMoyrOJIbHOM MBC C npHMLIM LC BHellIHHH yrOJI npH
BepmHHe A paBeH 120 0 , AB - AC = 9 CM. Haii.D;HTe AB.
4. H3 cepe HHbI D CTOpOHbI BC paBHocTopoHHero MBC rrpo-
Be eH nepneH)J;HKYJIHp DM K npHMoH AC. Haii.D;HTe AM,
eCJIH AB = 22 CM.
5. BepHo JIH yrBepm.D;eHHe: B paBH06e peHHoM TpeyrOJIbHHKe
6HcceKTpHcbI, rrpoBe eHHbIe H3 BepmHH OCHOBaHHH, paBHbI?
6. CYMMa KaTeTOB TpeyrOJIbllHK8 paBHa 11 CM, a O HH KaTeT
60JIbme ,D;pyroro Ha 5 CM. HaH,D;HTe IIJIOm;a.n;b TpeyrOJIhHHKa.
7. ,n:aH OCrrpoyroJIhHhIH M.BC, CD - Me 8Ha TpeyroJIbHHKa. Ha
CKOJIbKO qaCTeH eJI.HT IIJIOCKOCTh npHMble AB, BC, AC H CD?
Bapuanm 4
1. B paBH06e peHHoM MBC C OCHOBaHHeM AC npoBe eHa BhI-
COTa AD. HaH.D;HTe LCAD, eCJIH LB = 54 0
2. O HH H3 yrJIOB npHMoyrOJIbHOrO TpeyrOJIbHHKa paBeH 60°, a
CYMMa rHnOTeHY3bI H MeHbmero H3 KaTeTOB paBHa 40,5 CM.
HaH HTe MeHhmHH H3 KaTeTOB.
3. B rrpHMoyrOJIbHOM MBC C rrpHMbIM LB BHemHHH yrOJI npH
BepllIHHe C paBeH 120°, AC - BC = 7 CM. HaH)J;HTe AC.
4. H3 cepe HHbI S CTOpOHbI NP paBHOCTopOHHero MNP npo-
Be,n;eH rrepIIeH,n;HKYJIHp ST K rrpHMoH MP. HaH)J;HTe MT, ec-
JIH MN = 24 CM.
5. BepHo JIH YTBepm.D;eHHe: eCJIH KaTeT IIpHMoyrOJIbHOrO Tpe-
yrOJIbHHKa paBeH rrOJIOBHHe rHIIOTeHY3bI, TO yrOJI, JIema-
HH npOTHB 3Toro KaTeT8, paBeH 30 0 ?
6. CYMMa KaTeTOB TpeyrOJIbHHKa paBHa 9 CM, a O HH KaTeT
MeHbllIe pyroro Ha 3 CM. HaH)J;HTe rrJIOm; h TpeyrOJIbHHKa.
7. ,D;aH rrpHMoyrOJIbHbIH tV3CD, DE - 6HcceKTpHca TpeyrOJIb-
HHKa. Ha CKOJlbKO qaCTeH eJIHT IIJIOCKOCTb npHMbIe BC,
CD, BD H DE?
72
CaMOCTOHTeJIbHa.a pa60Ta NQ 35
Bapuanm 1
1. YroJI, npoTHBOJIema Hii OCHOBaHHIO paBH06e,ZJ;peHHoro Tpe-
yrOJIbHHKa, paBeH 120°. BblcoTa, rrpoBe,ZJ;eHHa.a K 60KOBOH
CTopOHe, paBHa 8 CM. H HTe OCHOBaHHe TpeyrOJIbHHKa.
2. B MBC AB = BC = 13,6 CM, BD - BbICOTa MBC, BD =
= 6,8 CM. HaH,ZJ;HTe LABC.
3. BbICOThI, npoBe;:J;eHHbIe K 60KOBbIM CTopOHaM AB HAC OCT-
poyrOJIbHOrO paBH06e,ZJ;peHHoro ABC, nepeceKaIOTCH B
TOtlKe M. Haii,ZJ;HTeyrJIbI MBC, eCJIH LBMC paBeH 160°.
4. BDICOTbI AD H BE MBC nepeceKalOTC.a B TOtlKe M. HaH,ZJ;H-
Te LAMB, eCJIH LBAC = 54°, LABC = 68°.
5. B PaBH06e,ZJ;peHHOM MBC C OCHOB8HHeM AC npOBe,ZJ;eHhI 6HC-
ceKTpHca AF H BbICOTa AH. Haii:,ZJ;HTe yrJIbI MHF, eCJIH
LABC = 108°.
6. CYMMa BYX CTOpOH npHMoyrOJIbHHKa ABCD paBHa 13 CM,
a o.n;Ha H3 3THX CTOpoH 60JIbme ,ZJ;pyroH Ha 3 CM. HaH)J;HTe
nJIOm;a.n;b MBC.
7. BepHo JIH YTBep.m,ZJ;eHHe: CYMMa ,l];BYX OCTpbIX yrJIOB rrpHMO-
yrOJIbHOrO TpeyrOJIbHHKa paBHa 90°?
Bapuaum 2
1. YroJI, npOTHBOJIejKa HH OCHOBaHHIO paBH06e peHHoro
TpeyrOJILHHKa, paBeH 120°. HaH,ZJ;HTe BbICOTY, npoBe,ZJ;eH-
HYIO K 60KOBOH CTopOHe, eCJIH OCHOBaHHe TpeyrOJIbHHKa
paBHO 14 CM.
2. B MBC AB = BC = 11,4 CM, BD - BbICOTa MBC, BD =
= 5,7 CM. Haii,ZJ;HTe CYMMY yrJIOB npH OCHOBaHHH AC.
3. BDICOTbI, npoBe.n;eHHbIe K 60KOBbIM CTopOHaM BC H CD OCT-
poyrOJIbHOrO paBHo6e,ZJ;peHHoro BCD, nepeceKaJOTCH B
TO'tlKe N. Haii,ZJ;HTe yrJIDI iV3CD, eCJIH LBND paBeH 140°.
4. BbICOTbI AD H BE MBC nepeceKalOTCH B TOqKe M. Haii,ZJ;H-
Te LAMB, eCJIH LBAC = 52°, LABC = 66°.
5. B paBHo6e,ZJ;peHHoM iV3CD C OCHOBaHHeM BD rrpOBe.n;eHbI
6HcceKTpHca BE H BbICOTa BH. Haii,ZJ;HTe yrJIbI iV3HE, eCJIH
LBCD = 104°.
6. CYMMa )J;BYX CTOpOH np.aMoyrOJIbHHKa ABCD paBHa 15 CM,
a o.n;Ha H3 3THX CTOpOH MeHbme ,ZJ;pyroH Ha 7 CM. Haii,ZJ;HTe
nJIO ap;L MBC.
7. BepHo JIB YTBepm.n;eHHe: eCJIH KaTeTbI O,ZJ;Horo npHMoyrOJIb-
Horo TpeyrOJIbHHKa COOTBeTCTBeHHO paBHbI KaTeTaM ,n;pyro-
ro, TO TaRHe TpeyrOJIbHHKH paBHhI?
CaMOCTOHTeJIhHaSl pa60Ta N2 35
73
Bapual£m 3
1. ¥roJl, rrpOTHBO.JIe.ma Hi1: OCHOBaHHIO p8.BHo6e peHHoro Tpe-
yrOJIhHHKa, paBeH 120°. BhIcoTa, rrpoBeAeHHaH K 60KOBOH
CTopOHe, paBHa 9 CM. HaHAHTe OCHOBaHHe TpeyrOJIhHHKa.
2. B MEC AB = BC = 15,2 CM, BD - BhICOTa MEC, BD =
= 7,6 CM. Haii,ZJ;HTe LABC.
3. BbICOTbl, rrpoBe,u;eHHbIe K 60KOBbIM CTopOHaM AB If AC OCT-
poyrOJIbHOrO paBHo6e.n;peHHoro ABC, nepeceKalOTCH B
TOllKe M. HaH HTe yrJIbI MBC, eC.JIH LBMC paBeH 150°.
4. BbICOTbI AD H BE MEC rrepeceK8IOTCH B TOllKe M. Haii,ZJ;H-
Te LBMD, eCJIH LBAC = 53°, LABC = 65°.
5. B paBHo6e peHHoM ABC C OCHOBaHHeM AC IIpOBe.n;eHbI
6HcceKTpHca AF If BbICOTa AH. Hai1.D;HTe yr .JIbl F, eCJIH
LABC = 112°.
6. CYMMa .D;BYX CTOpOH IIpsMoyrOJIhHHKa ABCD paBHa 12 CM,
a o,n;Ha If3 :3THX CTOpOH 60.JIbme ,lI;pyroH Ha 4 CM. HaH.n;HTe
n.JIO a.n;b MEC.
7. BepHo JIH YTBepm,n;eHHe: eCJIH rHIIOTeHY3a H OCTpbIH yrOJI
o,n;Horo IIpHMoyrOJIhHOrO TpeyrOJIhHHKa COOTBeTCTBeHHO
paBHbI rHIIOTeHY3e H OCTPOMY yr.JIY pyroro, TO TaKHe Tpe-
yrOJIbHHKH paBHbI?
Bapual£m 4
1. YroJI, npOTHBOJIema HH OCHOBaHHIO paBH06e peHHoro
TpeyrOJIhHIfKa, paBeH 120°. HaH,lI;HTe BhICOTY, rrpOBe,ll;eH-
HYIO K 60KOBOH CTopOHe, eC.JIH OCHOBaHHe TpeyrOJIbHHKa
paBHO 12 CM.
2. B MEC AB = BC = 9,8 CM, BD - BbICOTa MEC, BD =
4,9 CM. Hai1:,D;HTe CYMMY yrJIOB IIPH OCHOBaHHH AC.
3. BbICOThI, rrpoBe eHHhle K 60KOBbIM CTopOHaM BC If CD OCT-
poyrO.JIbHOrO paBHo6e,n;peHHoro BCD, nepeceKalOTCH B
TOQKe N. HaH HTe yrJIbI BCD, eCJIH LBND paBeH 130°.
4. BbICOTbI AD H BE MEC nepeceKalOTCH B TOllKe M. HaH,lI;H-
Te LAME, eCJIH LBAC = 55°, LABC = 69°.
5. B paBH06e peHHoM .6.BCD C OCHOBaHHeM BD rrpOBe.n;eHbI
6HcceKTpHca BE H BbICOTa BH. HaHAHTe yrJIbI .6.BHE, eCJIIf
LBCD = 96°.
6. CYMMa ,lI;BYX CTOpOH npHMoyrOJIbHHKa ABCD paBHa 16 CM,
a o.n;Ha H3 3THX CTOpOH MeHbme ,n;pyroH Ha 8 CM. HaH HTe
nJIO Mb MEC.
7. BepHo JIH YTBepm eHHe: eCJIH rHnoTeHyaa H KaTeT OAHoro
rrpHMoyro.JIhHOrO TpeyrOJIhHHKa COOTBeTCTBeHHO paBHhI rH-
rrOTeHY3e H KaTerry pyroro, TO TaKHe TpeyrOJIbHHKH paBHbl?
74
CaMOCTOHTeJIbHaSl pa60Ta N2 36
Bapuanm 1
1. BepHo JIH YTBepJK eHlie: KaTeT npSlMoyrOJIbHOrO TpeyrOJIL-
HHKa, JIemam;IIH npOTHB yr JIa B 30°, paBeH rrOJIOBHHe rHTIO-
TeHY3bI?
2. AB H CD - ,nllaMeTpbI oKpymHocTH C n;eHTpOM 0, AB1-CD,
TO Ka M JIemHT Ha pa Hyce OC, TO'tlKa E - Ha p Hyce OD,
MO = EO. CpaBHHTe AM H BE.
3. Ba rrpHMoyrOJIbHbIX TpeyrOJIhHIIKa MEC Ii MED HMelOT
06 ylO rllrroTeHyay AB II JIemaT no pa3HbIe CTOpOHhI OT Hee,
AD = BC. CpaBHHTe LCAB II LDBA.
4. MEC II MNS paBHbI, BC = NS H AB = MN. BepHO JIll YT-
Bepm.n;eHHe: BbICOTbI 3THX TpeyrOJIbHHKOB BD H NT paBHbI?
5. Ha CTopOHax AB H BC MEC B3S1TLI TO'tJKII D II E COOTBeTCT-
BeHHO. DK1-AC H EP J..AC, DK = EP, AB = 6 CM, LAnK =
= LP EC. HaH,lJ;IITe BC.
6. qepea cepe.n;HHY CTOpOHbI AB ABC npOBe eHa npHMaSl,
rrepneH.n;IIKYJISlpHaSl K AB, nepeceKalOru;aH BC B To'tJKe E.
BC = 24 CM, nepHMeTp AAEC paBeH 29 CM. Haii,lJ;HTe AC.
7. BepHo JIH YTBepm eHHe: KaTeT 60JIbme rllrroTeHyabI? .
Bapuanm 2
1. BepHo JIIi YTBepm,lJ;eHHe: KaTeT npSlMoyrOJIbHOrO TpeyrOJIb-
HHKa, JIeJKa HH npOTIIB yr JIa B 60°, paBeH nOJIOBIIHe rllno-
TeHY3bI?
2. BC H DE - .n;HaMeTpbI oKpymHocTH C n;eHTpOM 0, BC1-DE,
To'tJKa M JlemHT Ha pa,lJ;lIyce 0 D, TO'tJKa N - Ha PMHyce 0 E ,
OM = ON. CpaBHHTe BN Ii CM.
3. ,l];Ba npHMoyrOJIbHhIX TpeyrOJIbHIIKa MNP 11 NS IIMe-
lOT o6 ylO rllnoTeHyay MN Ii JIemaT no paaHbIe CTOpOHbI OT
Hee, MP = NS. CpaBHIITe LMNP H LNMS.
4. MEC H NS paBHbI, BC = NS H AB = MN. BepHO JIll yrBep-
m.n;eHHe: 6HCCeKTpHCbI 3THX TpeyrOJIbHHKOB BD H NT paBHLI?
5. Ha CTopOHax AB II BC MEC B3,HTLI TO'tlKH M H N COOTBeTCT-
BeHHO. MP 1-AC II N8UC, AM = CN, MB = 3 CM, LAMP =
= LCNS. Haii.n;HTe BN.
6. Qepe3 cepe.n;HHY CTOpOHbI AB ABC rrpOBe.n;eHa npSlMaH,
nepneH,lJ;HKYJISlpHaH K AB, nepeceKalOru;aSl BC B TO'tlKe N.
BC = 22 CM, rrepHMeTp MNC paBeH 28 CM. Haii.n;IITe AC.
7. BepHo JIll YTBepJK.n;eHHe: KaTeT MeHbme rIlIIoTeHY3bI?
C8MOCTOHTeJlhHaSl pa60T8 M 36
75
BapuaJtm 3
1. BepHo JIll YTBepm eHHe: eCJIIi KaTeT rrpSlMoyroJlhHoro Tpe-
yrOJIbHHKa paBeH nOJIOBHHe rHnOTeHY3bI, TO yrOJI, JIema-
HH npOTHB 3TOrO K8TeTa, paBeH 30°?
2. AB Ii CD - HaMeTphI oKpymHocTH C eHTpoM 0, AB-LCD,
To'tJKa M JIemliT Ha pa)J;liyce OC, To'tJKa E - Ha PMHyce aD,
MO = EO. CpaBHHTe LOAM H LOBE.
3. ABa npHMoyrOJIhHLIX TpeyrOJIhHliKa MEC Ii MED HMelOT
o6 yIO rHnoTeHY3Y AB H JIemaT no pa3HbIe CTOpOHhI OT Hee,
AD = BC. CpaBHHTe AC H BD.
4. MEC H NS paBHbI, BC = NS H AB = MN. BepHo JIll YT-
Bepm.n;eHHe: BhICOThI 3THX TpeyrOJIbHliKOB AD H MT paBHhI?
5. Ha CTopOHax AB H BC MEC B3S1ThI TO'tlKH D Ii E COOTBeTCT-
BeHHO. DK-LAC H EP -LAC, DK = EP, AB = 5 CM, LAnK =
= LPEC. HaHtJ;liTe BC.
6. "lJepe3 cepe)J;HHY CTOpOHhI AB MBC npOBe)];eHa npHMaH,
nepneH HKYJISlpHaSl K AB, nepeceKaIO aSl BC B TO'tlKe E.
BC = 26 CM, nepHMeTp MEC paBeH 30 CM. HaH HTe AC.
7. BepHo JIH YTBepmtJ;eHHe: B npSlMoyrOJIbHOM TpeyrOJIhHHKe
Ba npSlMhIX yr Jla?
BapuaKm 4
1. BepHo JIH YTBepm eHHe: eCJIH KaTeT npHMoyrOJIhHOrO Tpe-
yrOJIhHHKa paBeH nOJIOBHHe rHnOTeHY3hI, TO yrOJI, JIema-
ID;IIH npOTHB 3Toro KaTeTa, paBeH 60°?
2. BC H DE - HaMeTphI oKpymHocTH C eHTpOM 0, BCl.DE,
TO'tlKa M JIemHT H8 pap;Hyce aD, To'tJKa N - Ha pB.J];Hyce OE,
OM = ON. CpaBHliTe LOMC H LONB.
3. ABa npSlMoyrOJIhHbIX TpeyrOJIhHHKa t1MNP H MNS HMe-
lOT 06m;yIO rHnOTeHY3Y MN H JIemaT no pa3HbIe CTOpOHhI OT
Hee, MP = NS. CpaBHHTe PN H MS.
4. L1A..BC Ii f1MNS paBHhI, BC = NS H AB = MN. BepHO JIll YTBep-
eHHe: 6HcceKTpHchI 3THX TpeyroJIhHHKOB AD H MT PaBHhI?
5. Ha CTopOHax AB H BC L1A..BC B3S1ThI TOT.lKH M H N COOTBeTCT-
BeHHO. MP lAC Ii NSlAC, AM = CN, MB = 4 CM, LAMP =
= LCNS. HaH HTe BN.
6. Qepe3 cepe liHY CTOpOHbI AB f1ABC rrpOBetJ;eHa npHMaSl,
nepIIeH HKYJISlpHaH K AB, nepeceKalOIII;aH BC B TO'tlKe N.
BC = 21 CM, rrepHMeTpJiANCpaBeH 24 CM. HaH,ZJ;HTeAC.
7. BepHo JIll YTBepJKtJ;eHHe: B npHMoyrOJIbHOM TpeyrOJIhHHKe
,lJ;Ba OCTpbIX yr Jla ?
76
CaMOCTOHTeJIbHaH pa60Ta N2 37
BapuaKm 1
1.1f3 TO"tlKH K npHMoH npOBe eHbI nepneH HKYJI.Hp H HaKJIOH-
HaSI, CYMMa JIHH KOTOpbIX paBHa 15 CM, a pa3HOCTb ,l);JIHH
paBHa 3 CM. Haii HTe paCCTOHHHe OT TOqKH O npSIMoii.
2. B paBHOCTopOHHeM ABC npOBe.n;eHa 6HcceKTpHca AD.
PaccToSIHHe OT TOQKH D o npHMoH AC paBHO 8 CM. HaH H-
Te paCCTOSIHHe OT BepmHHhI A ));0 npHMoH BC.
3. CyMMa rHIIOTeHY3bI CE H KaTeTa CD npHMoyrOJIbHOrO CDE
paBHa 29 CM, a HX pa3HOCTb paBHa 5 CM. Haii.n;HTe JIHHY rH-
II OTeHY3bI.
4. BepHo JIH YTBepm,l);eHHe: B paBHo6e.n;peHHoM TpeyrOJIbHHKe
cepe HHa OCHOBaHHH paBHOY aJIeHa OT 60KOBbIX CTOpOH?
5. PaccToHHHe Mem y naparrJIeJIbHbIMH rrp5lMbIMH a H b paBHO
2 CM, a Mem.n;y napaJIJIeJIbHbIMH rrpSIMbIMIi a 11 C paBHO 7 CM.
HaH HTe paCCTOHHHe Mem y rrpHMbIMH b He.
6. IIp5lMbIe AB Ii CD rrapaJIJIeJIbHbI. Hai1: HTe paCCTOHHHe Me-
m,l);y AB H CD, eCJIH LADC = 30°, AD = 10 CM.
7. B TpeyrOJIhHHKe O,l];HH yrOJI B 3 pa3a 60JIbme Pyroro H Ha 40°
MeHbme TpeTbero. HaHAHTe 60JIbmHH yrOJI TpeyrOJIbHHKa.
BapuaKm 2
1. M3 TOqKH K npHMoH npOBe eHbI IIeprreH HKYJISIp H HaKJIOH-
HaH, CYMMa )J;JIHH KOTOpbIX paBHa 16 CM, a pa3HOCTL JIHH
paBHa 2 CM. Hai1:.n;HTe JIHHY HaKJIOHHoi1:.
2. B paBHOCTopOHHeM MNP rrpOBe eHa Me.n;HaHa PS. Pac-
CTOHHHe OT TOqKH S ));0 rrpHMoH MP paBHO 7 CM. HaH HTe
paCCTOHHHe OT BepmHHbI P ));0 npHMOH MN.
3. CYMMa KaTeTa AB H rHrrOTeHY3bI AC npSIMoyrOJIbHOrO MBC
paBHa 23 CM, a HX pa3HOCTb paBHa 3 CM. Haft));HTe paCCTOH-
HHe OT BepmHHbI A ));0 npHMoH BC.
4. BepHo JIll YTBepjK eHHe: neprreH HKYJIHp, rrpOBe.n;eHHbIH 113
TOqKH K rrpHMoi1:, MeHbme JII060H HaKJIOHHOH, rrpOBe,l];eH-
HOH H3 TOH me TOqKH K aToH npHMoH?
5. PaCCTOflHHe Mem,n;y rrapaJlJleJIbHLIMH npHMbIMH m II n paB-
HO 3 CM, a Mem y rrapaJIJIeJIbHbIMH rrpHMLIMH m H k paBHO
8 CM. HaH.n;HTe paCCTOSIHHe Mem));y rrpHMbIMH n H k.
6. IIpHMbIe AB H CD rrapaJIJIeJIbHbI. HaH HTe paCCTOHHHe Me-
m y AB H CD, eCJIH LBCD = 30°, BC = 12 CM.
7. B T})eyrOJIbHHKe O));HH yrOJI B 2 pa3a MeHbme pyroro II Ha 20°
60JIhme TpeTbero. HaH,lI;HTe MeHbillHH yrOJI TpeyrOJIbHHKa.
CaMOCTOHTeJIhHaH pa60Ta M 37
77
Bapuanm 3
1. M3 TO'tlKH K npHMoH npOBe eHhI nepneH HKYJIHp H HaKJIOH-
HaH, CYMMa .u;JIHH KOTOpbIX paBHa 1 7 CM, a paaHOCTb JIHH
paBHa 3 CM. Haii.n;HTe paCCTOHHHe OT TOtlKH O npHMoH.
2. B paBHOCTopOHHeM ABC rrpOBe.ZJ;eHa 6HcceKTpHca CD.
PaccToHHHe OT TO'tIKH D.n;o rrpHMoH AC paBHO 6 CM. HaH H-
Te paCCTOHHHe OT BeplIIHHhI C O npHMOH AB.
3. CYMMa rHnOTeHY3hI CE Ii KaTeTa CD npHMoyrOJIbHOrO CDE
paBHa 25 CM, a HX pa3HOCTb paBHa 7 CM. HaH.ZJ;HTe .ZJ;JIHHY rH-
IIOTeHY3bI.
4. BepHo JIM YTBepm.n;eHHe: Bce TO'tlKH Kam.n;OH H3 .n;BYX rrapaJI-
JIeJIhHbIX npSlMbIX paBHoy.n;aJIeHbI OT PyroH rrpHMoH?
5. PaCCTOHHHe MejK Y napaJIJIeJIbHhIMH rrpHMbIMH a H b paBHO
1 CM, a Mem.ZJ;Y rrapaJIJIeJIbHbIMH rrpHMhIMH a H C paBHO 8 CM.
Haii.ZJ;HTe paCCTOHHHe MejK.n;y npHMbIMH b He.
6. IIpHMhIeAB Ii CD napaJIJIeJIbHbI. HaH HTe paccToHHHe Me-
m.n;y AB H CD, eCJIH LADC = 30°, AD = 14 CM.
7. B TpeyrOJIbHHKe O.u;HH yrOJI B 2 paaa 60JIbme .ZJ;pyroro H Ha 30°
MeHbme TpeTbero. Haii.ZJ;HTe 60JIblliHH yrOJI TpeyrOJIbHHKa.
Bapuanm 4
1. H3 TOtlKH K npHMoH npOBe.n;eHbI neprreH HKYJIHp H HaKJIOH-
HaH, CYMMa .u;JIHH KOTOpLIX paBHa 14 CM, a paaHOCTb .n;JIHH
paBHa 2 CM. Haii,ZJ;HTe .ZJ;JIHHY HaKJIOHHoH.
2. B paBHOCTopOHHeM MNP npOBe eHa Me HaHa MS. Pac-
CTOHHHe OT TO'tIKH S .u;o rrpHMoH MP paBHO 9 CM. HaH.ZJ;HTe
paCCTOHHHe OT BepllIHHbI M .n;o npHMoH NP.
3. CYMMa KaTeTa AB H rHnoTeHyabI AC npHMoyrOJIbHOrO
MBC paBHa 27 CM, a HX paaHOCTb paBHa 5 CM. HaH HTe
paCCTOHHHe OT BeplIIHHbI A .ZJ;O npHMoH BC.
4. BepHo JIH YTBepm eHHe: no aaKOHY OTpameHHH CBeTa: na-
.ZJ;alOID;HH Ha npHMYIO JIyq II JIyq, oTpameHHbIH OT rrpHMoii,
COCTaBJIHIOT C .u;aHHoii rrpHMoii paBHhIe yrJIbI?
5. PaCCTO.HHHe Mem.u;y rrapaJIJIeJIbHbIMH rrpHMhIMH m H n paB-
HO 4 CM, a Mem y naparrJIeJIbHbIMH npHMbIMH m H k paBHO
7 CM. Haii.ZJ;HTe paCCTOSlHHe Mem y npHMbIMH n H k.
6. IIpHMbIe AB H CD napaJIJIeJIbHhI. Haii,ZJ;HTe paCCTOHHHe Me-
m.n;y AB H CD, eCJIH LBCD = 30°, BC = 16 CM.
7. B TpeyroJIbHHKe O.ZJ;HH yroJI B 3 pa3a MeHbllie pyroro H Ha 20°
60JIhlIIe TpeTbero. HaH HTe MeHblliHH yroJI TpeyroJIbHHKa.
78
CaMOCTOHTeJIbHaH pa60Ta N2 38
BapuaKm 1
1. MO.iKHO JIH rrOCTpOIlTb TpeyrOJIbHHK no BYM CTopOHaM II
yrJlY Me.iK Y HHMH?
2. MOjKHO JIH nOCTpoHTb TpeyrOJIbHHK no KaTeTY II rHnOTeHyae?
3. B paBH06e peHHoM MBC OCHOBaHHe AC = 28 CM, BHemHHH
yrOJI npH BepllIHHe B paBeH 60 C . HaH IITe paCCTOHHlIe OT
BepmHHbI C o rrpHMoH AB.
4. B fj,MNP LM = 70°, LN = 80°, NS - 6HcceKTpHca, PS =
= 8 CM. HaR HTe paCCTOHHHe OT TO"tlKH S O rrpHMOH MN.
5. qepea cepe HHY CTOpOHbI BC BC npOBe eHa npHMaSl,
neprreH,lJ;HKYJIHpHaH K BC, nepeceKalOm;aH AB B TO"tlKe D,
AB = 18 CM, rrepllMeTp futDC paBeH 26 CM. HaH IITe AC.
6. B npSlMoyrOJIbHOM MBC LC - rrpSlMoii, 6HcceKTpHcbI CD H
AE nepeceKalOTCSI B TOtIKe O. LAOC = 105°. HaH IITe OCT-
pLle yrJIbI AABC.
7. BepHo JIll YTBep.iK eHlIe: OCHOBaHHe paBHo6e.n;peHHoro Tpe-
yrOJIbHIIKa napaJIJIeJIbHO 6HcceKTpllce O Horo H3 BHemHHX
yrJIOB?
Bapu4Km 2
1. MomHo JIll nOCTpOHTb TpeyroJlbHHK no CTopOHe H ,n;BYM
npHJIemam;HM K HeH yr JIaM?
2. MomHo JIll rrOCTpOIlTb TpeyrOJIbHHK no .n;BYM KaTeTaM?
3. B paBH06e,lJ;peHHOM BCD OCHOBaHHe BD = 24 CM, BHem-
HHH yrOJI rrpH BepmllHe C paBeH 60°. HaH HTe paCCTOSlHHe
OT BepllIHHbI D o rrpHMoH BC.
4. B MBC LB = 80°, LC = 70 , BE - 6HcceKTpllca, AE = 6 CM.
HaH HTe paCCTOHHlIe OT TOQKH E o npSlMoH BC.
5. qepea cepe IIHY CTOpOHbI CD !1BCD rrpOBe,n;eHa rrpHMaH,
rreprreH,lJ;HKY JISlpHaH K CD, rrepeceKalOm;aSl BC B TOqKe E,
BC = 16 CM, nepHMeTp IV3DE paBeH 25 CM. HaH IITe BD.
6. B rrpHMoyrOJIbHOM dARC LA - rrpHMoii, 6HcceKTpllcbI AD
II CE nepeceKalOTcSI B TOqKe o. LAOe = 125 0 . Haii,n;HTe OCT-
pble yrJIbI LlABC.
7. BepHo JIH YTBep.iK.n;eHHe: eCJIH KaTeT 11: rrpHJIemaru;IIR K He-
MY OCTpbIH yrOJI O Horo TpeyrOJIbHHKa COOTBeTCTBeHHO
paBHbI KaTeTY H rrpllJIemam;eMY K HeMY OCTpoMY yrJlY py-
roro, TO TaKlIe TpeyrOJIbHHKH paBHbI?
CaMOCTOHTeJIbHaH pa60Ta NQ 38
79
BapuaKm 3
1. MomHo JIH nOCTpOHTb TpeyrOJIbHHK no TpeM cTopoHaM?
2. MomHo JIH nOCTpOHTb TpeyrOJIbHHK no rHnoTeHyae H IIPH-
JIema eMY K HeH OCTPOMY yrJIY?
3. B paBHo6e peHHoM MBC OCHOBaHHe AC = 26 CM, BHemHHR
yrOJI rrpH BepmHHe B paBeH 60 0 . HaH,D;HTe paCCTOHHHe OT
BepmHHbI C ,n;o npHMoH AB.
4. B MNP LM = 70°, LN = 80°, NS - 6HcceKTpHca, PS =
= 10 CM. HaH,ZJ;HTe paCCTOHHHe OT TO'tIKH S O npHMOR MN.
5. 1Jepea cepe,ZJ;HHY CTOpOHbI BC MBC npOBe eHa npHMaH,
nepneH.n;HKYJIHpHaH K BC, IIepeceKalO aH AB B TO'tlKe D,
AB= 15 CM, IIepHMeTpMDCpaBeH 22 CM. HaH,n;HTeAC.
6. B npHMoyrOJIbHOM MBC LC - rrpHMoH, 6HcceKTpHcLI CD H
AE rrepeceKalOTCSI B TOT.lKe O. LAOe = 115°. HaH,ZJ;HTe OCT-
pbIe yrJIbI MBC.
7. BepHo JIH YTBepm.n;eHHe: eCJIH rHIIoTeHyaa H OCTpbIH yrOJI
O,l];HOrO npHMoyrOJIbHOrO TpeyrOJIbHHKa COOTBeTCTBeHHO
paBHbI rHnOTeHY3e H OCTPOMY yrJIY ,ZJ;pyroro, TO TaKHe Tpe-
yrOJIbHHKH paBHbI?
BapuaKm 4
1. MomHo JIH nOCTpOHTb TpeyrOJIbHHK rro TpeM yrJIaM?
2. MomHO JIM rrOCTpOHTh TpeyrOJILHHK no KaTeTY H rrpHJIema-
m;eMY K HeMY OCTPOMY yrJIY?
3. B paBH06e,n;peHHoM CD OCHOBaHHe BD = 22 CM, BHem-
HHH yrOJI npH BepmHHe C paBeH 60°. Haii,ZJ;HTe paCCTOHHHe
OT BepmHHhI D ,n;o npHMoH BC.
4. B MBC LB = 80°, LC = 70°, BE - 6HcceKTpHca, AE = 12 CM.
Ha:ti,l];HTe paCCTOHHHe OT TO'tlKH E .n;o npHMoH BC.
5. "tJepea cepe,ZJ;HHY CTOpOHbI CD BCD rrpOBe.n;eHa npHMa.H,
IIepneH,l];HKYJIHpHa.a: K CD, IIepeceKaID aH BC B TO'tlKe E,
BC = 18 CM, nepHMeTp BDE paBeH 24 CM. Haii HTe BD.
6. B IIpHMOyrOJIbHOM MBC LA - rrpHMoH, 6HcceKTpHcbI AD
H CE nepeceKaIDTCH B TOQKe o. LAOC = 100°. Haii,l];HTe OCT-
pble yrJIbI MBC.
7. BepHo JIH YTBepm,ZJ;eHHe: eCJIH rHnOTeHY3a II KaTeT O,l];HOrO
rrpSiMoyrOJIbHOrO TpeyrOJIbHHKa COOTBeTCTBeHHO paBHbI rH-
IIOTeHY3e H KaTeTY .n;pyroro, TO TaKHe TpeyrOJIbHMKH paBHbI?
80
OTBeTLI
CaMOCTOHTeJlbHaH pa60Ta N 1
lJ BapHaHT 1 BapHaBT 2 BapHaHT 3 BapHaRT 4
1 6 10 15 21
2 3 1 2 1
3 5 6 5 6
4 )J;a ,IJ;a ,JJ;a )J;a
5 3 2 3 2
6 a ,JJ;a 6 10
7 a ,JJ;a ,IJ;a ,lJ;a
CaMOCTOHTeJIbHaH pa60Ta N2 2
C BapHaHT 1 BapHaRT 2 BapnaHT 3 BapHaRT 4
1 10 15 21 28
2 5 4 5 4
3 8 10 8 10
4 )J;a Aa ,JJ;a a
5 )J;a ,lJ;a ,JJ;a ,na
6 HeT HeT HeT HeT
7 15 25 15 25
CaMOCTOHTeJIbHaH pa60Ta N 3
0 BapHaHT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 ,na )J;a a ,JJ;a
2 6 4 10 8
3 )J;a ,lJ;a Aa ,na
4 10 8 11 10
5 12 16 12 16
6 12 12 4 4
7 HeT HeT HeT HeT
OTBeThI
81
CaMOCTOJlTeJILHaJi pa60Ta N!l 4
C BapaaBT 1 BapHaHT 2 BapaaHT 3 BapHaBT 4
1 12 4 12 4
2 )];a )];a a a
3 Aa a )];a ,IJ;a
4 8 13 8 13
5 4 3 4 3
6 Aa a ,IJ;a ,IJ:a
7 16 12 16 20
CaMOCTOBTeJlbHaJi pa60Ta N!l 5
0 BapHaHT 1 BapaaBT 2 BapaaJIT 3 BapaaHT 4
1 4 10 4 10
2 ,lJ;a Aa ,IJ;a ,IJ;a
3 3 2 3 2
4 22,5 6 45 0 18 0 36 0
5 ,IJ;a ,IJ;a ,IJ;a Aa
6 2 CM 7 CM 5 CM 10 CM
7 40 CM 2 24 CM 2 28 CM2 18 CM 2
CaMOCTOJlTeJlbHaJi pa60Ta N!l 6
0 BapaaHT 1 Bapa80T 2 BapHaBT 3 BapaaHT 4
1 ON OB OM OA
2 ,IJ;a ,IJ;a BeT HeT
3 MC = AB BN = AC NC = AP BM = AK
4 HeT HeT a ,IJ;a
5 )];a )J;a ,IJ;a ,IJ;a
6 84 0 30° 80° 20 6
7 10 0 70 0 25° 90°
82
OTBeTLI
CaMOCTOJlTeJIhHaH pa60Ta N 7
;l BapaaHT 1 BapaalfT 2 BapHaHT 3 BapaaoT 4
1 1 CM; 1 7 CM 2 CM; 16 CM 1 CM; 15 CM 2 CM; 12 CM
2 HeT ,IJ;a HeT ,IJ;a
3 3 CM 9 CM 2 CM 11 CM
4 2 CM 8 CM 16 CM 6 CM
5 0,75x 2 0,75b 2
_ox -.b
3 3
6 )];a ,IJ:a ,D;a ,IJ;a
7 10 6 10 6
CaMOCTOJlTeJIhHaB pa60Ta N 8
[ BapHaHT 1 BapaaoT 2 BapaaHT 3 BapaaHT 4
1 13 CM 5 CM 10 CM 4 CM
2 ,IJ:a HeT ,IJ:a HeT
3 x:2 n:2 m: 2 k:2
4 7 5 4 3
_.x -.n -.m -.k
8 6 5 4
5 17 CM 14 CM 10 CM 13 CM
6 20 CM 36 CM 2 28 CM 2
64 CM
7 2 TO'lKH 2 TOtlKH 2 TOqKH 2 TOtJKH
CaMOCTOJlTeJIhHa.H pa60Ta N2 9
[j BapHaHT 1 BapaaoT 2 ! BapHaHT 3 BapHaHT 4
1 48 31 c 49 0 33°
2 84 0 72 0 105 0 84 c
3 HeT ,IJ;a HeT ,D;a
4 ,l];a HeT ,IJ;a I HeT
5 120 0 100 0 140 0 130 0
6 36° 108 0 54° 72 0
9 45' 45 45' 45"
OTBeThI
83
CaMOCTOJlTeJIhHaJl pa60Ta Ng 10
D BapHaHT 1 BapHaBT 2 BapHaHT 3 BapHaHT 4
1 20 0 80 0 30 0 100 0
2 HeT ,IJ;a HeT ,n;a
3 72° 80 0 77 0 96°
4 120 0 30 0 120 0 30 0
5 15 0 90 0 10 Q 80 0
6 135 0 140 0 145° 150°
7 ,n;a HeT ,IJ;a HeT
CaMOCTOJlTeJlhHa.a: pa60Ta N2 11
0 BapHaBT 1 BapHaHT 2 BapaaHT 3 BapxaHT 4
1 65 0 , 65 0 , 115 0 , 75 0 , 75 0 , 105 0 , 55°, 55°, 125°, 45°, 45 c , 135 0 ,
115 0 105 0 125 0 135 0
2 50 0 120 0 30 0 110"
3 100° 72 0 135 0 40°
4 J];a ,IJ;a ,IJ;a ,n;a
5 ,n;a HeT HeT ,IJ;a
6 60° 40 0 30 0 10"
7 40 0 100 0 50° 80 0
CaMoCTowreJlhHaJi pa60Ta N 12
D BapHaHT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 HeT HeT HeT HeT
2 72° 108 0 60° 150 0
3 HeT HeT HeT HeT
4 10° 15 0 20 0 25°
5 8 8 12 12
6 HeT ,IJ;a HeT ,n;a
7 40° 20° 15° 45°
84
OTBeTItI
CaMOCTOHTeJILHaB pa60Ta N2 13
C BapHaHT 1 BapHaBT 2 BapHaHT 3 BapHaHT 4
1 )];a ,IJ;a )J:a )];a
2 90 0 180 0 90" 180"
3 ,IJ;a .l];a ,ZJ;a )];a
4 50° 40° 30° 20 0
5 18° 36" 30° 60°
6 10 12 6 8
7 100 35° 85 0 30"
CaMOCTOBTeJIbHaS pa60Ta N 14
0 BapHaHT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 35" 25" 40 30°
2 6 CM HJIH 10 CM 6 CM HJIH 8 CM 4 CM HJIH 8 CM 6 CM HJIM 12 CM
3 )J:a HeT ,ZJ;a HeT
4 HeT ,D;a ,IJ;a ,IJ;a
5 HeT HeT HeT HeT
6 2 CM 2 CM 3 CM 3 CM
7 8 8 8 8
CaMOCTOJlTeJIbHaH pa60Ta N 15
0 BapHaHT 1 BapHaHT 2 BapHaHT 3 ! BapHaHT 4
1 ,D;a ,IJ;a ,IJ;a i ,IJ;a
2 HeT HeT HeT HeT
3 40 0 30 c 50" 20 0
4 45 CM 36 CM 33 CM 39 CM
5 I ,I(a ,L(a ,L(a ,n;a
6 3 CM 3 CM 4 CM 4 CM
7 I 110? 100 e 115 0 105"
I
OTBeTLI
85
CaMOCTOJlTeJlhHaJi pa60Ta N 16
0 BapHaHT 1 BapHaHT 2 BapaaHT 3 BapHaHT 4
1 50° 80 0 60 0 90 0
2 48° 44" 46° 52°
3 ¥ I' JIbl paBHhI Y I' JIb! paBHLI ¥ r JILl paBHb! ¥I'JILI paBHhI
4 O.n;HH O.n;HH O));HH O HH
5 HeT ,[{a HeT .IJ;a
6 .IJ;a HeT ,IJ;a HeT
7 HeT a HeT .IJ;a
CaMOCTOJlTeJlhHaH pa60Ta N2 1 7
-
0 BapaaHT 1 BapaaHT 2 BapaaHT 3 BapBaHT 4
1 .l.(a ,IJ;a ,IJ;a ,D;a
2 .l.(a ,IJ;a ,D;a )J:a
3 ,IJ;a ,IJ;a ,D;a ,IJ;a
4 .IJ;a HeT .l.(a HeT
5 AH AC AM AH
6 0 0 0 0
7 160 0 140 0 120° 100 0
CaMOCTOJlTeJlhH8.JI pa60Ta Ng 18
D BapaaHT 1 BapaaHT 2 BapHaHT 3 BapHaHT 4
1 8 CM 7 CM 5 CM 12 CM
2 11 CM 8 CM 10 CM 6 CM
3 6 CM 4 CM 8 CM 5 CM
4 )J:a ,IJ;a ,IJ;a .l.(a
5 .l.(a ,IJ;a .IJ;a )J:a
6 ,IJ;a J];a .IJ;a )J:a
7 50°, 50° t 80 ; 40°,40 0 , 100°; 55°, 55°, 70°; 45°,45°, 90°;
70°, 70° t 40° 80°, 80°, 20° 65°, 65°, 50° 75°, 75°, 30 0
86
OTBeThI
CaMOCTOJlTeJIhHaJi pa60Ta N!! 19
r BapHaHT 1 BapaaHT 2 BapHaHT 3 BapHaHT 4
i
1 12 CM 11 CM 14 CM 15 CM
2 BD = CD BD = BE AB = AC DC = CE
3 .l];a )J:a )J;a .l];a
4 )J:a HeT HeT )J:a
5 HeT HeT HeT HeT
6 )];a ,IJ;a HeT )J:a
7 HeT )];a HeT HeT
CaMOCTOHTeJIbHaJi pa60Ta N!! 20
0 BapHaHT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 ,lJ;a HeT HeT )J:a
2 )];a ,IJ;a )J:a )];a
3 YrJIbI paBHbI YrJIbI paBHbI YrJIhI paBHhI YrJIbI paBHhI
4 70° 60" 75" 65 0
5 .l];a )J;a ,IJ;a )J:a
6 40° 90< 20° 80"
7 )];a .l];a )J:a ,n;a
CaMOCTOJlTeJIbHaJi pa60Ta N 21
L BapHaHT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 ,IJ;a ,IJ;a .D;a ,n;a
2 90< 45° 45" 90"
3 Orpe3KH paBHbI Orpe3KH p8.BHbI OTpe3KH paBHbI OTpe3KH paBHbI
4 .D;a ,D;a )];a .l];a
5 ,lJ;a .D;a .u;a )J;a
6 BD = AC BD > EF AB < CE CE == DF
7 HeT HeT HeT HeT
OTBeTId
87
CaMOCTOJlTeJIhHaJl pa60Ta N! 22
0 BapHaBT 1 BapnaHT 2 BapHaHT 3 BapHaHT 4
1 42 CM 41 CM 44 CM 43 CM
2 AB = AC MN = MP CA = AB PM = MN
3 ,IJ;a a ,IJ;a ,IJ;a
4 HeT HeT HeT HeT
5 ,IJ;a Aa ,IJ;a ,IJ;a
6 P8BHo6e.zu>eHHbIH Pa3HOCTOpoHHl-lii Pa3HOCTOpOHHHH PUHo6eJU)eHHhIH
TYTIoyrOJIbHhIH IIpHMOyroJIbHbIH llpHMOyroJIbHhIH TYIIoyrOJIbHbIH
7 )J;a )];a ,IJ;a I )J;a
I
CaMOCTOJITeJIbHaH pa60Ta N2 23
I
[J i BapHaBT 1 BapHaHT 2 BapHaHT 3 BapnaHT 4
I
1 i Herr HeT HeT Herr
I
2 I a ,IJ;a )J;a )J;a
I
3 HeT HeT HeT HeT
4 ,IJ;a )];a )J;a ,IJ;a
5 )];a ,IJ;a ,IJ;a )];a
6 )];a ,IJ;a a ,IJ;a
7 6 CM HJIH 10 CM 6 CM HJIH 14 CM 4 CM HJIH 8 CM 8 CM HJIH 16 CM
CaMOCTOHTeJIhHaJl pa60Ta N 24
0 BapHaBT 1 BapnaHT 2 BapHaHT 3 BapHaHT 4
1 ,IJ;a ,IJ;a ,n;a ,IJ;a
2 HeT HeT HeT HeT
3 ,IJ;a HeT ,IJ;a HeT
4 ,IJ;a ,IJ;a ,IJ;a )J;a
5 HeT ,IJ;a ,IJ;a HeT
6 40" 60° 20° 30"
7 10 9 10 9
88
OTBeTLI
CaMOCTOHTeJIhHaH pa60Ta N 25
l BapHaHT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 .D;a )J:a )];a .n;a
2 J];a ,n;a )];a )J:a
3 80° BD = BE 60° BD = BE
4 ,LJ;a )];a .n;a ,LJ;a
5 ,n;a HeT )J:a HeT
6 125<> 120 0 130 0 115 0
7 15 CM 12 CM 16 CM 6 CM
CaMOCTOJlTeJIhHaJl pa60Ta N 26
[ BRpHRHT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 ,LJ;a HeT ,n;a HeT
2 HeT ,lJ;a HeT )];a
3 .L(a )];a HeT HeT
4 72° 120 0 36 0 20 0
5 60° 60 0 120 0 120 0
6 90 0 45 0 90° 45 0
7 40 50' 30 0 60 0
CaMOCTOHTeJIhHaH pa60Ta N 27
0 BapHRoT 1 BapHaHT 2 BapHaHT 3 BapHRoT 4
1 HeT HeT HeT HeT
2 2 HJIH 3 1 IIJIH 2 2 HJIH 3 1 HJIIf 2
3 110 0 H 110° 70 0 H 70 0 120 0 II 120 0 80" If 80 0
4 "tIeTblpe yrJIa no qeTLIpe yr JIa no "tIeTblpe yr JIa no qeTblpe yr JIa no
80 0 II 'tJ:eTblpe 50 c II qeTblpe 70" H qeTblpe 60" If qeTbIpe
yrJIa no 100 0 yr JIa no 130 0 yrJIa no 110 0 yr JIa no 120 0
5 75° Ii 105 0 80° H 100 0 65" H 115 70< If 110"
6 18" 26 c 8" 4 0
7 )la .L(a ,IJ;a )];a
OTBeThI
89
CaMOCTOJlTeJIbHaH pa60Ta N2 28
0 BapHaHT 1 BapH8BT 2 B8pHaHT 3 BapHaBT 4
1 1 1 1 1
2 qeTbIpe yr Jla no qeTbIpe yr JIa no qeTbIpe yr Jla no "CleTbIpe yr JIa no
45 0 If tleTblpe 55 0 H "CleTbIpe 65 0 H qeTbIpe 75 0 H "CleTbIpe
vrJla no 135 0 vrJIa no 125 0 yrJla no 115 0 yrJIa no 105 0
3 Aa HeT Aa HeT
4 .n;a HeT Aa HeT
5 HeT )];a .n;a .n;a
6 110 0 120 0 120 0 130 0
7 55 0 165° 35° 105 0
CaMOCTOHTeJlbHaH pa60Ta Ng 29
] BapHaBT 1 BapHaBT 2 BapHaBT 3 BapHaBT 4
1 )];a )];a .n;a )];a
2 )];a HeT .n;a HeT
3 )];a .n;a HeT )];a
4 tleThIpe yr JIa no tleTLIpe yr Jla no qeTbIpe yr Jla no qeThipe yr Jla no
30 0 " 't£eTLIpe 36 0 H tleThipe 45 0 B tleTLIpe 60 0 H tleThlpe
vr JIa no 150 0 vrJla no 144 0 yrJIa no 135 0 vrJla no 120 0
5 HeT HeT ,na )J;a
6 8 CM 20 CM 16 CM 25 CM
7 6 CM 5 CM 5 CM 4 CM
CaMOCTOJlTeJIbHaH pa60Ta N, 30
0 BapH8HT 1 BapnaBT 2 BapHaBT 3 BapH8BT 4
1 60 0 20 0 90 0 60 0
2 40° 80 0 20 0 100 0
3 )J;a HeT ,na HeT
4 HeT .n;a HeT )J;a
5 103 0 101 0 104 0 102 0
6 58 0 HJIH 64 0 44 0 "JIB 68 0 59 0 HJIH 62 0 48 0 OJIH 66 0
7 36 CM 34 CM 40 CM 32 CM
90
OTBeTLI
CaMOCTO.HTeJIhHaS pa60Ta N2 31
J BapHaHT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 72° 36° 80 0 140 0
2 44°,44 0 . 92" 41 0 t 41", 98° 46 0 , 46 0 t 88° 42°. 42°, 96°
3 108 0 102 0 99 c llr
4 HeT HeT HeT HeT
5 a )];a HeT HeT
6 }];a ,IJ;a )J:a .l];a
7 12 CM 16 CM 10 CM 20 CM
CaMOCTOBTeJIhHaS pa60Ta N 32
[ BapHaHT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 20° 40 0 20(> 40°
2 45", 45°, 90 0 36°, 36 0 t 108° 30 0 , 30°, 120° 20 0 , 20 c , 140 0
3 HeT .l];a HeT )J;a
4 .l];a HeT )];a HeT
5 HeT )];a HeT HeT
6 16 CM 20 CM 21 CM 18 CM
7 )J:a )];a ,IJ;a .l];a
CaMOCTOBTenbHas pa60Ta N2 33
[J BapHaBT 1 BapHaHT 2 BapHauT 3 BapHaHT 4
1 BC CD NP DE
2 HeT )];a HeT ,[{a
3 5 CM 12 CM 3 CM 10 CM
4 12 CM 10 CM 14 CM 16 CM
5 29 CM; 29 CM 24 CM; 24 CM 31 CM; 31 CM 28 CM; 28 CM
6 .l];a HeT )];a HeT
7 to? 80 0 30° 70°
OTBeThI
91
CaMOCTOHTeJlhHaH pa60Ta N2 34
0 BapHaBT 1 BapHaHT 2 BapH8HT 3 BapHaBT 4
1 26° 28° 29° 27°
2 21 CM 12,5 CM 23 CM 13,5 CM
3 16 CM 12 CM 18 CM 14 CM
4 13,5 CM 15 CM 16,5 CM 18 CM
5 Aa )J:a ,[{a )J;a
6 2 24 CM 2 2 2
6 CM 12 CM 9 CM
7 10 10 10 10
CaMOCTOBTeJJbHaH pa60Ta N2 35
0 BapHaHT 1 BapHaHT 2 BapHaBT 3 BapHaHT 4
1 16 CM 7 CM 18 CM 6 CM
2 120° 60° 120 0 60
3 20°; 80°; 80° 40°; 70°; 70° 30°; 75°; 75° 50°; 65°; 65°
4 122 118 0 62° 56°
5 36°; 54°; 90 0 38°; 52°; 90° 34°; 56°; 90° 42°; 48°; 90°
6 20 CM 2 2 2 2
22 CM 16 CM 24 CM
7 ,IJ;a Aa ,1]; a ,IJ;a
CaMOCTOJlTeJIhHaJi pa60Ta N 36
[1 BapHaBT 1 BapHaHT 2 BapHaHT 3 BapHaHT 4
1 a HeT ,IJ;a HeT
2 AM = BE EN = CM YrJIbI paBHbI Y r JIhI paBHbI
3 YrJIhI paBHLI Y r JIbI paBHhI AC = BD PN = MS
4 Aa a ,l];a ,IJ;a
I
5 6 CM 3 CM 5 CM 4 CM
6 5 CM 6 CM 4 CM 3 CM
7 HeT ,IJ;a HeT ,IJ;a
92
OTBeThI
CaMOCTOBTeJIhHaH pa60Ta N 37
r- BapuaHT 1 BapHauT 2 BapHaHT 3 BapHaBT 4
L..
1 6 CM 9 CM 7 CM 8 CM
2 16 CM 14 CM 12 CM 18 CM
3 17 CM 10 CM 16 CM 11 CM
4 Aa ,IJ;a a .l];a
5 5 CM HJlH 9 CM 5 CM HJIH 11 CM 7 CM HJHI 9 CM 3 CM HJIH 11 CM
6 5 CM 6 CM 7 CM 8 CM
7 100(> 30" 90" 20°
CaMOCTOHTeJlhHaH pa60Ta N 38
n BapHaHT 1 BapHaBT 2 BapHaHT 3 BapHaHT 4
1 Aa ,L\a a HeT
2 HeT ,IJ;a HeT .l];a
3 14 CM 12 CM 13 CM 11 CM
4 4 CM 3 CM 5 CM 6 CM
5 8 CM 9 CM 7 CM 6 CM
6 30° Ii 60° 20° Ii 70° 40° H 50° 20" H 70°
7 )J:a a ,L\a ,L\a
co,n:EPmAHHE
II petJ;HCJIOBHe. . . . . . . . .. ........ 3
CaMoCTOHTeJIbHaH pa60Ta N2 1 . . . . . . . . . . 4
CaMoCTOHTeJIbHaH pa60Ta N2 2 . . . . . . 6
CaMoCTOHTeJIbHaH pa60Ta NQ 3 . . .. ...... 8
CaMoCTOHTeJIbHaH pa60Ta NQ 4.. ...... 10
CaMoCTOHTeJIbHaH pa60Ta N 5. . . . . . . . . . . . 12
CaMoCTOHTeJIbHaa pa60Ta N 6. . . . . . . . . . . . 14
CaMoCTOHTeJIbHaa pa60Ta M 7. . . . . . 16
CaMoCTOHTeJIhHaa pa60Ta M 8. . . . . . . . . . . . 18
CaMoCTOHTeJILHaa pa60Ta M 9.. ..... . . . 20
CaMOCTOHTeJIhHaa pa60Ta NQ 10 . . . . . . . . . . . 22
CaMOCTOHTeJIbHaH pa60Ta N2 11 . . . . . . . . . . . 24
CaMOCTOHTeJIbHaa pa60Ta N 12 . . . . . . . 26
CaMOCTOSlTeJIbHaSl pa.6oTa M 13 . . . . . . . . . . . 28
CaMoCTOHTeJIbHaH pa60Ta N2 14 . . . . . . . . . . . 30
CaMoCTOHTeJIbHaa pa60Ta N2 15 . . . .. .... 32
CaMoCTOHTeJIbHaH pa60Ta N2 16 . . . . . . . . . . . 34
CaMoCTOaTeJIbHaH pa60Ta M 1 7 . . . . . 36
CaMoCTOaTeJIbHaH pa60Ta NQ 18 . . . . . . .. . 38
CaMOCTOHTeJIhHaH pa60Ta NQ 19 . . . . . . . . . . . 40
CaMoCTOHTeJIbHaH pa60Ta NQ 20 . . . . . . . . . . . 42
CaMoCTOHTeJIbHaH pa60Ta N2 21 .... .. . 44
CaMoCTOaTeJIbHaH pa60Ta M 22 .... . 46
CaMoCTOHTeJIbHaH pa60Ta M 23 . . . 48
CaMoCTOSlTeJIbHaH pa60Ta NQ 24 . . . . .. ... 50
CaMoCTOHTeJIbHaH pa60Ta N2 25 . . . . . . . . . . . 52
CaMoCTOHTeJIbHaH pa60Ta N2 26 ....... 54
CaMoCTOHTeJIbHaH pa60Ta N2 27 ...... . 56
CaMoCTOHTeJIbHaH pa60Ta N2 28 .... . 58
CaMoCTOaTeJIbHaH pa60Ta M 29 . 60
CaMOCTOaTeJIbHaH pa60Ta N2 30 . 62
CaMoCTOHTeJIhHaH pa60Ta N2 31 . . . . .. . 64
CaMoCTOHTeJIbHaH pa60Ta N2 32 ..... e . 66
CaMoCTOHTeJIhHaSI pa60Ta N2 33 . .. ...... 68
CaMoCTOHTeJIbHaH pa60Ta N2 34 . . . . . . . .. 70
CaMoCTOHTeJIbHaSl pa60Ta N2 35 .... ... 72
CaMoCTOHTeJIbHaa pa60Ta M 36 . . . . . . . .. 74
CaMoCTOHTeJIbHaH pa60Ta N2 37 . . . ...... 76
CaMoCTOHTeJIbHaH pa60Ta N2 38 . .. ...... 78
OTBeTbI. . . . . . . . . . . . .. ...... 80