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2004

BBK 22.151 SI72
II43
3ameepa:HCeno MintcmepcmeoM oceimu i nay1eu Y1epa'inu
(JlHCT MiHicTepcTBa ocBiTH i HayKH YKpai:HH N.! 1/ 11- 3801BiA17.09.2001)
IlpoAaata llpHMipBHKiB UhOl'O BHAaHHll
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HaB'laJibHe BHARHHH
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rEOMETPUI
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IliAPY'IHHK AJIH 7-9 1rnaciB
aaraJibHOOCBiTHiX HRB'IRJibHHX 3RKJIRAiB
3am8epa:JK:eno MinicmepcmBOM ocBimu i nay1w Y1epa'inu
BiAIIOBiAaJibHHH aa BHIIYCR IO. 0. Kop6yrn.
XyAo:llrne o<t>opMJieHHH A. A. 3iH'leHRa.
IIi,11nHCaHO ,!10 APYKY 10.09.04. <l>opMaT 60x90/16. Ilanip ocjlc. rapHiTypa lllKiJibHa.
APYK ocpc. YM. ,11pyK. apK. 15. YM. cpap60Bi,116. 15,5. 06JI.-BH.zl. apK. 15,48.
Tupa>K 10 000 np. 3aM. N~ 4-492.
YBU •IIlKOJillP•.
02094, KHi:B, Byn. Ceprit:HKa, 18.
CBi,llOUTBO AK N.! 360 Bi,ll 14.03.2001 p.
BHAPYKOBaHO Ha BAT •Binon:epKiBCbKa KHH>KKOBa cpa6pHKa.,
09117, M. Bina I.{epKBa, ByJI. Jlec11 Kyp6aca, 4.
npasa aBTOpie Ta B11AaBH114rea 3ax11~eHi 3aKOHOM YKpa'iH11 «npo asropcbKe npaso
i cyMi>KHi npasa» BiA 23.12.1993 p.
APYKOBaHe Konit0saHHl'l KH11r11 a6o 'Ii 4acT11H11, 6YAb-RKi iHwi KOHTpacpaKTHi B11AaHHl'l
Tl'lrHyrb 3a co6ot0 ei,Qnoei,c.anbHiCTb 3riAHO 3i CT. 44. n. 1.3 4boro 3aKOHy.
Iloropt:JIOB 0. B.
II43 reoMeTpix: IIJiaHiMeTpix: IliAPY'I. AJIH 7-9 KJI. 3araJibHO-
OCBiT. HRB'I. 3RKJI.- 7-Me BHA.- K.: lIIROJIHp, 2004. - 240 C.
ISBN 966-7117-65-0
ISBN 966-711 7-65-0
BBK 22.1511172
© 0 . B. floropenos. 1994
© 0. B. floropenos, 2003.
3i 3MiH8MH
© BHA8BHHL{TBO •lllKOJillp• ,
xy11o>KKe O<jlopMJieHHR, 2004

7 KJlaC
§ 1. OCHOBHI BJIACTHBOCTI
HAHIIPOCTIWHX rEOMETPH1UIHX CZ-IrYP
1. rEOMETPlfllHI fl»lrYPH
I'eoMerpia - ~e uayKa npo BJiacTHBOCTi reoMeT}>H'IHHX cpiryp.
CnoBo .reoMeTpi1u - rpe~Ke, y nepeKJI&Ai ua yRpaiHci.Ry MOBY
oaua11ae •aeMneMipcTBO•. T&Ka uaasa ayMoBJieua aacTOCYB&HHHM
reoMeTpii AO BHMipIOBa.Hb Ha Mic~eBOCTi.
IlpHKJI8P,H reoMeTpH'IHHX cpiryp: TJ>HK}'THHK, KBap,paT, KOJIO
(M8Jl. 1).
reoMeTpH'IHi cpirypH 6yeaIOTb P,OCHTb pi3HOM8HiTHi. q8CTHH8
6yp,b-HKOi reoMeTpH'IHOi cpirypH e reoMeTpH'IHOIO cpirypoIO. 06'ep,-
H8HHH KIDKOX reoMeTpH'IHHX cpiryp e auosy reo11eTpH'IHOIO cpiry-
poIO. Ha MaJIIOHKY 2 cpirypa a.ni.Ba cKJiap,aeTLCH a TPHKYTHHKa i
Tpi.ox KBap,paTie, a cpirypa cnpaBa - a Kona i 'l&CTHIDI KOJia. Komuy
reo11eTpH11Hy 4>iJ"¥PY MO>KH8 YRBHTH CKJiap,eHOIO 3 TO'IOK.
DOMan. 1
Man. 2

7 KJiac
EeKJii'A - .n;aem.o-
rpe~bKHH y'leHHil
(JU CT. 'AO H. e.)
4
reoMeTpiR illHpoKO 3aCTOCOBYGTLCR Ha
npaKTHaj. Ii: Tpe6a 3HaTH i po6iTHHKy, i
iHmeHepy, i apxiTeKTopy, i XYAOlKHHKy.
0AHHM CJIOBOM, reoMeTpiio c~ 3HaTH BCiM.
reoMM'piR, ~O BHB'laGTbCR y illKOJii,
HaaHBaGTLCR eBKJ1iAoB010, aa iM'RM AaBHbo-
rpeD,bKoro B'leHoro EBKJiiAa, RKHH CTBOPHB
noci6HHK 3 MaTeMaTHKH niA Ha3BOJO
•Ha11aJia•. TpHBaJIHii 11ac reoMeTpiIO BH-
B'laJIH aa ~GIO KHHlKKOIO.
Mu no'IHHaGMo BHB'leHHR reoMeTpii a
nJiaHiMe-rpii. llJr.aHiMerpia - ~e poaAiJI
reoMeTpii, B RKOMY BHB'laIOTLCR <PirypH Ha
IlJIO~Hi.
2. TO'IKA I IIP.HMA
0cHOBHHMH reoMeTpH'IHHMH <l>irypaMH Ha IlJIO~HHi e TO'l.H:a i
npsut.a. To'IKH npHiiHRTO noaHa'laTH BeJIHKHMH JiaTHHCbKHMH
6yKBaMH A, B, C, D, .... IIpaMi noaHa•1aIOTLCR MaJIHMH JiaTHH-
CbKHMH 6YKBaMH a, b, c' d, ....
Ha MaJIJOHKY 3 BH 6a'IHTe TO'IKY A i npaMy a.
IIpaMa HecKiH'leHHa. Ha MaJIIOHKY ao6pamyeMo JIHme 11acTHHY
npaMoi, aJie yRBJIRGMo ii Heo6MemeHo npOAOBlKeHOIO B o6HABa
60KH.
IIOAHBiTbCR Ha MaJIIOHOK 4. BH 6a11HTe npaMi a, b i TO'IKH A,
B, C. To'IKH A i C Jr.e:JKarb Ha npaMiii a. MomHa cKaaaTH TaKOlK,
~o TO'IKH A i C HaJr.e31Carb npaMiii a, a6o npRMa a npoxoiJurb
11epea TO'IKH A i C.
a
Man. 3 Man. 4

§ 1. Ocuoeui BJI&CTHBOCTi ua8npOCTimux reoMeTpH'IHHX cl>iryp 5
To'IKa B nemHTb Ha rrpsMtii: b. BoHa He nemHTb Ha rrpHMiii a.
To'IKa C nemHTb i Ha rrpHMtii: a, i Ha rrpHMiH b. IlpHMi a i b neperu-
H.alOTbC.R. B TO'I~ C. To11Ka C 6 ro11.1eo10 neperuny rrpHMHX a i b.
Ha MaJIIOHKY 5 BH 6a'IHTe, HK aa p;orr0Moro10 niHtii:KH MomHa
rro6yp;yuaTH rrpsMy, ~o rrpoxop;HTb qepea p;ui TO'IKH A i B.
Man. 5
0 c H 0 B H H M H B JI a c T H B 0 c T SI M H HaJie)KHOCTi TO'IOK i
IIpSIMHX Ha IIJIO~HHi HaaBeMO TaKi BJiaCTHBOCTi:
I. HKa 6 He 6yll4 nplUl&a, icHyHJT'b TO"Ku, UfO Ha11.e:11eaT'b ifirl
np1u•iu, i ro"JCJi, UfO He Ha11.e•aT'b iu.
qepe3 6yiJ'b-JUri iJ6i TO"ICU M.031&"4 npot1ecTU npaMy i TiJl.'blCU
OiJHy.
IlpHMY MOmea rroaeatiaTH ,IJ;BOMa TO'IKaMH, ~o nemaTb Ha HiH.
HarrpHKna,11;, rrpsMy a Ha MaJIIORKY 4 MOmHa rroaHa'IHTH AC, a nps-
MY b-BC.
3 a ,II; a 'I a (3) 1
• liu MO)KYTb p;Bi rrpsMi rrepeTHHaTHCJI
y ,11;uox TO'IKax? Bi,IJ;rrouip;L rroscHiTL.
Po au' .a a a H H s. HK6H p;Bi IIpHMi MaJIH p;ui TO'IKH rrepe·
THHy, TO qepea ~ TO'IKH rrpoxop;HJIH 6 JJ;Bi rrpsMi. A ~e HeMOlK-
JIHBO, OCKiJibKH 11epea p;Bi TO'IKH MOmHa rrpoBeCTH TiJibKH OAHY
IIp.RMy. 0Tme, p;Bi rrpHMi He MO»tYTb MaTH ,IJ;BOX TO'IOK rrepe-
- THHy.
3. BIWI30K
Ilop;HBiTbCH Ha MaJIIOHOK 6. BH 6a'IHTe rrpsMy a i TPH TO'IKH
A, B, C Ha ~ rrpsMiii. To'UCa B nemHTb Mi.JK TO'IKaMH A i C, BOHa
po3ai;uie TO'IKH A i C. MomHa TaKom cKaaaTH, ~o TO'IKH A i C ne-
I t{HCJIO B .ll;Y)KK8X 03H8'18€ HOMep 3~aqi B CIIHCKY 3~8'1,
naseii;eeux y KiHI~i naparpact>a.

7 KJiac 6
MaJI:. 6 Man. 7
maTi. no pi3Hi 6oH:u Bi.A TO'IKH B. To'IKH Bi C JiemaTi. no oiJun 6iH:
BiA TO'IKH A, BOBH He pooAiJIBIOTi.ca TO'IKOIO A. To'IKH A i B JiemaTi.
no OAHH 6iK Bi.A TO'IKH c.
BiiJpi3KOM Ha3HBa6T'bCB 11acTHHa npaMo'i, mca cKJiaAaeTi.cs a
ycix TO'IOK ~e'i npBMOl, ~o JI8)1(8Tb Mim ABOMB AaHHMH i"i TO'IKaMH.
IJ;i TO'IKH Ha3HBaIOTbCB KiHi.µiMU eiiJpi31Ca. BiApiaoK noaHa'laIOTb,
aanHcyIO'IH iioro KiH~. KoJIH roBOpaTi. a6o nHmYTb oiApiaoK AB•,
TO MaIOTL Ha yBaai BiApiaoK 3 KiB::u;sMH B TO'IKaX A i B.
Ha MaJIIOHKY 7 6a'IHMO BiApiaoK AB. BiH e qacTHHOIO npaMo'i
AB. Il;IO 11acTHBY npaMo'i BHAi.neHo mHpHOIO JiiHieio. To'IK8. X npa-
Mo'i JiemHTb Mim TO'IKaMH A i B, TOMY BOHa HanemHTi. BiApi3Ky
AB. To'IKa Y He JiemHTL Mim TO'IKaMH A i B, TOMY BOHa He HaJiemHTL
Bi.Api3Ky AB.
0 c H 0 B H 0 IO B JI a c T H B i c T IO p03M~eHH$1 TO'IOK Ha npa-
Miii Ha3BeMO TBKY BJiaCTHBiCTb:
I I. 3 rp11ox ro'Wx Ha npiudu oiJHa i riJJ11rcu o8Ha Jie:11&Ur11 Mi:w
OBOlllQ. iHWUMU.
4. BHMIPIOBAHHH BIAJ>I3KIB
,lJ;JIB BHMipIOB8HH$1 BiApi3KiB KOpHCTYIOT'bC$1 piaHHMH BHMipIO-
B&JibHHMH iHcTpYMeHTaMH. HaiinpocTimHM T&KHM i.HcTpyMeHTOM
e JiiHiAKa a noAiJIKaMH Ha Hi.A. Ha M&JIIOHKY 8 BiApiaoK AB AOPiB-
RJOe 10 CM, BiApiaoK AC AOpiBHIOe 6 CM, a eiApiaoK BC AOPiBHIOe
4 CM. ,ll;OB)KHHa BiApiaKa AB AOPiBHIOe cyMi AOB)KHH eiApiaKiB
AC i BC.
0 c H 0 B H H M H B JI a c T H B 0 c T $1 M H BHMipIOBaHH$1 BiApia-
KiB H83BeMO TaKi BJI&CTHBOCTi:

§ 1. OcHoBHi BJiftCTHBOCTi HaiirrpocTimHx reoMeTpH'lHHX cl>iryp 7
A [ B
MaJI. 8
III. Ko:11eHuu BiiJpiaoK Mae neBHY iJ08:11tuHy, 6i".,,"'Y Bio "Y""·
,lfo831&UHa BiiJpia""' iJopitJHH>e cyxi 0083/eUH 'UJCTUH, Ha siKi BiH poa-
6UBaer'bCsi 6yiJ'b-1'1C010 UOZO T01'1C010.
~e 00Ha11ae, w;o KOJIH Ha eiJJ;piaKy AB eaaTH 6yJJ;L-aKy TO'IKY C,
TO AOB)KHHa BiApi3Ka AB AOPieHI06 cyMi AOB)KHH ei)l;pi3Kie AC
i BC. ,IJ;oe)KHHY eiJJ;piaKa AB HaaHeaIOTL TaKo)K Biocrann10 Mi)K
TO'IKaMH A i B.
3 a A a 11 a (9). TpH TO'IKH A, B, C Jie)KaTL Ha OAHiii npa-
Miii. Bi,loMo, w;o AB = 4,3 CM, AC = 7,5 CM, BC = 3,2 cM.
lJH MO)Ke To11Ka A Jie)K&TH Mi)K TO'IKaMH B i C? lJH MO)Ke To11-
Ka C Jie)KaTH Mi)K TO'IKaMH Ai B? .HKa a TpLox TO'IOK A, B, C
Jie)KHTL Mi)K ABOM& iHmHMH?
p 0 3 B. $1 3 a H H a. .HKw;o TO'IKa A Jie)KHTL Mi)K TO'IKaMH
B i C, TO aa enacTHBicTIO BHMip10eaHHa ei,lpi3Kie Mae 6yTH
AB + AC = BC. Ane 4,3 + 7,5 =I= 3,2. 0T)Ke, TO'IK& A He
ne)KHTL Mi>K TO'IK&MH B i C•
.HKw;o TO'IKa C Jie)KHTL Mi)K TO'IK&MH A i B, TO Mae 6yTH
AC + BC = AB. Ane 7,5 + 3,2 =I= 4,3. OT)Ke, TO'IKa C He
Jie)KHTL Mi)K TO'IK&MH A i B.
3 TpLox TO'IOK Ha npaMiii A, B, C OAHa ne)KHTL Mi)K ABOMa
iHWHMH. OT)Ke, ~e10 TO'IKOIO e TO'IKa B.
5. IIIBIIJIOID;BHB
lloAHBiTLca Ha MaJIIOHOK 9. llpaMa a poa6HBae nnow;HHY Ha ABi
nieIIJIOUJ;HHH. ~e po36HTT$1 Mae TaKy BJiaCTHBiCTL. .HKw;o KiH~
$1KOrO-He6yAL Bi,lpi3Ka HaJie)KaTL OJJ;Hiii niBIIJIOUJ;HHi, TO ei)l;pi30K
He nepeTHHae npa:My• .HKw;o KiHu;i BiApiaKa HaJie)KaTL piaHHM
nieIIJIOUJ;HHaM, TO ei,lpiaOK nepeTHHae npaMy.
Ha MaJIIOHKY 9 TO'IKH A i B ne)KaTL e OAHiii a niennow;HH, Ha aKi
npa:Ma a poa6ueae IIJIOUJ;HHy. ToMy BiApiaoK AB He nepeTHHae npa-
MY a. To'IKH C i D Jie)KaTL y piaHHX niennow;HHax. ToMy eiApiaoK
CD nepeTHHae npaMy a.

7 KJiac 8
Man. 9
0 c H 0 B H 0 IO B JI a c T H B i c T IO poaMim,eHHH TO'IOK BiAHOCHO
npHMOi Ha IlJIOlll;HHi H83BeMo TaKy BJiaCTHBiCTL:
W. HpRMa poa6usae 1U011fUHY Ha 8si niBn.1io11fliHu.
3 a A a 'I a (17). ,D;8Ho npHMY i TPH TO'IKH A, B, C, w;o He
JiemaTi. Ha Hiii. BiAoMo, w;o siApiaoK AB nepeTHHae npsMy, a
BiApiaoK AC He nepeTHHae ii. 't!H nepeTHHae npsMy BiApiaoK
BC? BiAilOBiAL IlOHCHiTL.
p 0 3 s' H 3 a H H H. Ilpa:Ma poa6HB86 IlJIOw;HHY Ha ABi niB-
IlJIOw;HHH (MaJI. 10). To'IKa A HaJiemHTL oAHiii a HHX. BiApiaoK
AC He neperHHae npa:My. 0Tme, TO'IKa C JiemHTL y Tiii caMiii
niBnJIOw;HHi, w;o ii TO'IKa A.
BiApiaoK AB nepeTHHae npsMy. 3Ha'IHTL, TO'IKa B JiemHTL
B iHmiii nisnJiow;HHi. TaKHM 'IHHOM, TO'IKH B i C JiemaTL
y piaHHx uisuJIOw;HHax. A ~e OOHa11ae, w;o siApiaoK BC nepe-
THHae A8HY DPHMY.
6. IlIBilP.HMA
3aAa'la (20). .IJ:aHo upsMy a i TO'lKH A, X, Y, Z Ha Hitt
(MaJI. 11).BiAOMO, w;o TO'IKH x i y JiemaTL uo OAHH 6iK BiA
TO'lKH A, TO'lKH x i z Tem JiemaTL DO OAHH 6iK BiA TO'lKH A.
MaJI. 10 MaJI. 11

§ 1. 0CHOBHi BJiaCTHBOCTi Haii:npocTiIUHX reoMeTpH'IHHX cl>iryp 9
HK poaMiw;eHi TO'IKH Y i Z BiAHOCHO TO'IKH A : no OAHH 6iK,
'IH no pi3Hi 6oKH? IIOSicHiTh BiAnoBiAh.
Po 3 B' H 3 a H H H. IIpoBeAeMo qepea TO''IKY A AOBiJihHY
npHMY b, Bi,l.MiHHY BiA a. BoHa po3i6'e nJIOIIJ;HHY Ha ABi niB-
nJIOIIJ;HHH. 0AHiii 3 HHX HaJie:>KHTh TO'IKa x. y Tiii caMiii niB-
nJIOIIJ;HHi JiemaTh TO'IKH Y i Z, 60 Bi,l.pi3KH XY i XZ He nepe-
THHaIOTh npHMy b. OcKiJihKH TO'IKH x i z Jie:>KaTh B OAHiH
niBnJIOIIJ;HHi, TO BiApiaoK YZ He nepeTHHae npHMY b, a oT:me,
He MiCTHTh TO'IKY A. To6To TO'IKH Yi Z Jie:maTh no OAHH 6iK
BiA TO'IKH A.
llienpa.M.010 a6o npo.M.eHe.M. Ha3HBaIOTh qacTHHY npHMOl, HKa
CKJia).a6ThCH 3 ycix TO'IOK ajsi: npHMOl, w;o JiemaTb no OAHH 6iK BiA
).aHOl Ha Hiii TO'IKH. u;a TO'IKa Ha3HB86ThC.R llO'UlT1'06010 TO'l/COIO
niBnpaMoi:. Pi3Hi niBnpaMi OAHis1 ii Tisi: caMoi' npaMoi' i3 cniJihHOIO
no'laTKOBOIO TO'IKOIO H83HBaIOThC.R iJon06H.R.JtbHU.M.U.
IIiBnpaMi TaK caMo, .RK i npaMi, no3Ha'laIOThC.R MaJIHMH JiaTHH-
ChKHMH 6yKBaMH. Mo:mHa no3Ha'laTH niBnpJ1My ABOMa TO'IKaMH:
notJ:aTKOBOIO i w;e HKOIO-He6y,l.h TO'IKOIO, HKa HaJie:>KHTh niBnpaMiii.
1Iot1aTKOBY TO'IKY cTaBJI.RTh Ha nepmoMy Micaj. HanpHKJiaA,
niBnpaMy, HaBe).eHy :>KHPHOIO JiiHi6IO Ha M8JIIOHKY 12, MO:>KHa
n03H8'1HTH AB.
a A 8
MaJI. 12
3 a A a q a (22). Ha Bi,l.pi3Ky AB B3.RTO TO'IKY C. CepeA
niBnpHMHX AB, AC, CA i CB Ha3BiTh napy niBnpHMHX,
w;o 36iraIOThC.R, i napy AOnOBHHJihHHX niBnpaMHX. Bi,l.noBiAh
nO.RCHiTh.
P o 3 B' H 3 a H Ha (MaJI. 13). ,lJ;aHi niBnpaMi MaIOTh no-
'IRTKOBOIO TO'IKOIO a6o TO'IKY A, a6o TO'IKY C.
Po3rJiaHeMo cnoqaTKY niBnpaMi a no'laTKOBOIO TO'IKOIO
A (niBnpaMi AB i AC). To'IKa C Jie:>KHTh Mi:>K TO'IKRMH Ai B,
ocKiJihKH 3a yMOBOIO 3aAa'li BOHa HaJie:>KHTh Bi,l.pi3Ky AB.
0Tme, TO'IKa A He Jie:>KHTh Mi:>K TO'IKaMH Bi C, T06To TO'IKH
Bi C JiemaTh no OAHH 6iK Bi). TO'IKH A. ToMy niBnpHMi AB
i AC 36iraIOThCH.

7 KJIRC 10
MaJI. 13
PoarmrneMo Tenep nisnpHMi a no11aTKOBOIO TO'IKOIO C (nis-
npHMi CA i CB). To11Ka C poa,zdn:He TO'IKH A i B. ToMy TO'IKH A
i B He MO)KYTL HaJiemaTH O,ll;HIB nisnpHMIB, OTme, nisnp.11Mi
CA i CB ,ll;OilOBH$1JILHi.
7. KYT
KyroM HaausaeTLCH cl>irypa, HKa cKJia,n;aeTbCSI a TO'IKH - 6ep-
wunu Kyra i ,ll;BOX pi3HHX nisnpSIMHX, w;o BHXO,ll;SITL i3 ~iei TO'IKH,-
cropin Kyra.
Ha MaJIIOHKY 14 BH 6a11uTe KYT a sepmHHOIO 0 i cTopoHaMH
a, b. KyT noaHa11aIOTL a6o iioro sepmHHOIO, a6o iioro cTopoHaMH,
aoo aanHCYIOTL TpH TO'IKH: BepmHHY i ,ll;Bi TO'IKH, Ill;O Jie:>KaTL Ha
cropoHax KyTa. CJioBo •KYT• iHo,n;i aaMiHSIIOTL aHaKoM L. KYT Ha
MaJIIOHKY 14 MomHa nooHa'IHTH TpLoMa cnoco6aMu: L 0, L (ab),
L AOB. Y TpeTLOMY cnoco6i noaHa11eHHS1 KyTa 6yKsa, w;o noaHa'lae
sepm'HHY KyTa, CTaBHTLC.11 BCepe,n;HHi.
HKw;o cTopoHH KyTa e ,n;onoBHSIJILHHMH nisnpSIMHMH o,n;Hiei np.11-
Moi, TO KYT HaausaIOTL poazopnyruM. Ha MaJIIOHKY 15 au 6a'IHTe
poaropHyTuii KYT a sepmHHOIO 0 i cTopoHaMH OA i OB.
a
b
0 8
MaJI. 14 MaJI. 15 MaJI. 16

§ 1. Ocaosai BJiacTHBOCTi aaii:npocTiWHx reoMeTPH'IHHX «l>iryp 11
By,11;eMo roBopHTH, m;o npoMiHb npoxoiJurb Mi:HC cTopoHaMH
,ll;aHoro KyTa, HKIIl;O BiH BHXO,ll;HTb 3 H:oro BeplllHHH i nepeTHHa6
HKHH-He6y,11;L Bi,ll;piaoK a KiHD;HMH Ha H:oro cTopoHax. Ha MaJIIOH-
KY 16 npoMiHb c npOXO,ll;HTb Mim CTOpOHaMH KyTa (ab), OCKiJibKH
BiH BHXO,ll;HTb a BepmHHH KyTa (ab) i nepeTHHae Bi,ll;piaoK AB a KiH-
D;HMH Ha H:oro CTOpOHax.
,lJ;JIH poaropHyToro KyTa BBama6MO, m;o 6y,ll;b-HKHH npoMiHb,
HKHH BHXO,ll;HTb 3 H:or:o eeplllHHH i Bi,ll;MiHHHH Bi,ll; iioro cropiH, npo-
XO,ll;HTb Mim CTOpOHaMH KyTa.
KyTH BHMipIOIOTbCH rpa.zi;ycaMH aa .zi;onoMoroIO TpaHcnopTHpa.
Ha MaJIIOHKY 17 KYT (ab) .zi;opiBHIOe 120°. IliBnpHMa c npoxo,zi;HTb
Mim cTopoHaMH KYTa (ab). KyT (ac) .zi;opiBHIOe 90°, a KYT (be)
.zi;opiBHI06: 30°. KyT (ab) .zi;opiBHIOe cyMi KyTiB (ac) i (be).
0 c H 0 B H H M H B JI a c T H B 0 c T H M H BHMipIOBaHHJI KYTiB
Ha3BeMO TaKi BJiaCTHBOCTi:
V. Ko31CHU.U 'It.yr Mae nesHy zpaiJycHy 111iipy, 6ial>w.y siiJ Hya11..
PoazopHyru.4 'It.YT iJopi6HKH! 180°. rpaiJycHa xipa N.yTa iJopi6H10e
cyxi zpaiJycHux xip N.yri6, Ha 11.N.i siH poa6UtJaenc11. 6yiJb-ll.N.UX
npollU!HeX, UfO npoxoiJurb xi.31C 4ozo cropoHaXu.
I.J;e 03Hattae, m;o KOJIH npoMiHb c npOXO,ll;HTb Mim cropoHaMH
KyTa (ab), TO KYT (ab) ,11;opiBHI06 cyMi KyTiB (ac) i (be).
3 a ,!I; a 'I a 25. qH MO)Ke npoMiHb c npOXO,ll;HTH Mim CTOpo-
HaMH KyTa (ab), J1Km;o L (ac) = 30°, L (cb) = 80°, L (ab)=
= 50°?
Po a e' JI a au H JI. HKm;o npoMiHb c npoxo,zi;HTb Mim cTo-
pouaMH KyTa (ab), TO aa BJI8CTHBiCTIO BHMipIOBaHHJI KyTiB
Mae 6yTH L (ac) + L (cb) = L (ab). AJie 30° + 80° =F 50°.
OTme, npoMiHb c He Mome npoxo,zi;HTH Mim cTopoHaMH
KyTa (ab).
MaJI. 17

7 KJiac 12
8. BI,D;KJIA,D;AHH.H BI,D;PI3KIB I KYTIB
Ha MaJIIOHKY 18 noKa3aHo, HK 3a ,n;on0Moro10 JiiHiHKH Ha niB-
npHMiH a 3 noqaTKOBOIO TO'lKOIO A MOJKHa Bi,ll;KJiaCTH Bi,n;pi3oK
,n;aHOl ,ll;OBJKHHH (3 CM).
! / a
MaJI. 18
Ilo,n;HBiTbCH Ha MaJIIOHOK 19. IliBnpHMa a, npO,ll;OBJKeHa 3a no-
qaTKOBY TO'lKY A, po36HBas nJIO~HHY Ha ,n;Bi niBnJio~HHH. Ha
MaJIIOHKY noKa3aHO, HK 3a ,ll;OIIOMOrOIO TpaHCIIOPTHpa Bi,ll;KJiaCTH
Bi,n; niBnpHMoi a y BepxHio rriBilJIO~HHY KYT 3 ,n;aHOIO rpa,n;ycHoIO
MipOIO (60°).
0 c H 0 B H H M H B JI a c T H B 0 c T ff M H Bi,ll;KJia,n;aHHH Bi,n;pia-
KiB i KyTiB HaaBeMO TaKi BJiaCTHBOCTi:
VI. Ha 6yiJb-1uciu niBnpaMill Bia ii no11aT1to8o'i To•ucu Mo:NCna
BiiJJCJr.acTU BiiJpi301C iJaHO'i iJ08:11CUHU i TillblCU OiJUH.
VII. Bia 6yiJb-1'1CO'i niBnp1u,o'i y aany nienllOUfUHY MO:llCKa eiiJ-
JCJr.acmu Kym a iJaHOKJ zpaiJycKoKJ MipoKJ, MeKw.OKJ aa 180°, i
miJr.blCU oiJUK.
/'f' 3 a A a q a (30). Ha npoMeHi AB Bi,n;KJia,n;eHo Bi,n;piaoK AC,
V MeHIIIHH aa Bi,n;piaoK AB. .HKa a Tpbox TO'lOK A, Bi C JieJKHTb
MiJK ,ll;BOMa iHIIIHMH? IloHCHiTb Bi,n;noBi,ll;b.
MaJI. 19

§ 1. OcHOBHi BJiaCTHBOCTi Hai1:npocTiwHx reoMeTpH'IHHX <liiryp 13
p 0 3 B' H 3 a H H H (MaJI, 20).
OcKiJibKH TO'lKH B i C Jie:>KaTb Ha oAHiii
ni.JmpHMiii a no'laTK0~010 TO'IKOIO A,
TO BOHH He poa,a;iJIHIOTbCH TO'IKOIO A,
TOOTO TO'IKa A He Jie:>KHTb Mi:>K TO'IKa-
MH Bi c.
qH MO:>Ke TO'IKa B Jie:>KaTH Mi:>K
TO'IKaMH A i C? HK6H BOHa Jie:>KaJia
Mi:>K TO'IKaMH A i C, TO 6yJIO 6 AB +
+BC= AC. AJie ~e HeMO:>KJIHBo,
Man. 20
ocKiJibKH aa YMOBOIO Bi.ApiaoK AC MeHmHii aa Bi.a;piaoK AB.
0Tme, TO'IKa B He Jie:>KHTb Mim TO'IKaMH A i C.
3 Tpbox TO'IOK A, B, C OJJ;Ha Jie:>KHTb Mi:>K ABOMa iHIIIHMH.
ToMy TO'IKa C Jie:>KHTb Mim TO'IKaMH A i B.
9. TPHKYTHHK
TpuKyrnuJCoM HaaHBaGTbCH <Pirypa, HKa cKJia)J;aGTbCH a Tpbox
TO'IOK, ~o He Jie:>KaTb Ha O,ll;HiH npHMiH, i TpbOX Bi.ApiaKiB, HKi
nonapHo cnony'laIOTb Itj. TO'IKH. To'IKH Ha3HBaIOTbCH eepwunaMu
TPHKYTHHKa, a BiJJ;pi3KH - :Horo CTOpoHaMu.
Ha MaJIIOHKY 21 DH 6atIHTe TPHKYTHHK a eepmHHaMH A, B, Ci
CTOpoHaMH AB, BC, AC. TpHKYTHHK noaHa11aGTbCH iioro eepIIIHHa-
MH. 3aMiCTb cnoBa •TPHKYTHHK• iHOAi B:>KHBaIOTb aHaK t:,.. Ha-
npHKna.a;, TPHKYTHHK Ha MaJIIOHKY 21 noaHa'laeTbCH TaK: £:,.ABC.
KyroM rpuKyrnuKa ABC npH eepmHHi A H83HBaGTbCH KyT,
YTBOpeHHii nienpHMHMH AB i AC. TaK caMo oaHa11a10TbCH KYTH
TPHKYTHHKa npH eepmHHax Bi C.
A
,ll;Ba Bi.Api3KH H83HB8IOTbCH pi6HUMU, HK~O BOHH MaIOTb O)J;H8KO-
B
[ A
Man. 21
[
B A,
Man. 22
B,

7 KJiaC 14
BY AOB)l(HHy. ,IJ;sa KYTH Ha3HBIUOTbC$1 piBHUMU, SIKID;O BOHH MaIOTL
OAHaKOBY KYTOBY Mipy B rpap;ycax.
TplJKyTHHKH Ha3HBIUOTbCSI piBHUMU, SIK~o B HHX sip;noBip;Hi
CTOpOHH piBHi i Bip;noBiAHi KYTH piBHi. IlpH ~OMY BiAllOBi/J;Hi
KYTH MaIOTL JieJKaTH npoTH Bip;noBi/J;HHX CTOpiH.
Ha Ma.JIIOHKY 22 BH 6aqHTe p;Ba piBHHX TPHKYTHHKH ABC i
A1B1C1. y HHX
AB= A1Bi, AC= A1Ci. BC= B1Ci.
LA= L A1, LB= L Bi. LC= L Ci.
PiBHi sip;piaKH Ha Ma.JIIOHKy, SIK npaBHJIO, n00Haqa10TL OAHie10,
ABOMa a6o TPLOMa pHcKaMH, a piBHi KYTH - OAHieIO, ABOMa a6o
TPLOMa AY)l(KaMH.
,lJ;JISI noaHaqeHHSI piBHOCTi TPHKYTHHKiB KOpHcTyIOTLCSI 3BHqaii-
HHM 3HaKOM piBHoCTi =. 3anHc 6. ABC= 6. AiBiCi qHTaeTLCSI
TaK: •TPHKYTHHK ABC p;opiBHIOe TPHKYTHHKY A iB1C i•. IlpH ~LOMY
Mae 3HaqeHHSI nop$1p;OK aanHcy sepDIHH TPHKYTHHKa. PiBHiCTL
6. ABC= 6. AiB1C1 ooHaqae, ~o LA= L Ai, LB=
= L Bi, .... A pisHicTL 6. ABC= 6. BiAiC1 ooHaqae B)l(e aoeciM
iHme: L A = L Bi, L B = L Ai ....
3 a A a q a (38). TpHKYTHHKH ABC i PQR piBHi. Bip;o-
MO, ~o CTOpoHa AB p;opiBHIOG 10 M, a KYT C p;opiBHIOe 90°.
1IoMy p;opiBHIOIOTL CTopoHa PQ i KYT R? Bip;noeip;L no-
SICHiTL.
Po a e' SI a a H H 11. OcKiJILKH TPHKYTHHKH ABC i PQR
piBHi, TO AB = PQ, L C = L R. OT)l(e, PQ = 10 M, LR =
= 90°.
10. ICHYBAHHH TPHKYTHHKA, ~O f);OPIBHIOO f);AHOMY
Hexaii MaeMo TPHKYTHHK ABC i npoMiHL a (MaJI. 23, a). Ilepe-
MiCTHMO TPHKYTHHK ABC TaK, ~o6 iioro eepmHHa A a6irJiacs a
noqaTKOM npoMeHH a, sepmHHa B 6yJia Ha npoMeHi a, a eepDIHHa C
OilHHHJiacg y p;aHiii niBllJIO~HHi Bip;HOCHO npoMeHH a Ta iioro npo-
AOB)l(0HH$1. BepmHHH Hamoro TPHKYTHHKa y ~MY HOBOMy llOJIO-
)l(eHHi noaHaqHMO Ai. Bi. C1 (MaJI. 23, 6).
TpUICYTHUK AiBiC1 aopiBHJOe rpu-,cyrnu-,cy ABC.
lcHyBaHHH TPHKYTHHKa A1BiCi, ~o p;opiBHIOG TPHKYTHHKY ABC
i poaMi~eHHii 3a3HaqeHHM cnoco6oM BiAHOCHO p;aHoro npoMeHH a,

§ 1. OcHOBHi BJiaCTHBOCTi HaiinpocTiUIHX reo:r.i:eTpH'IHHX cl>iryp 15
a
[
A B
a) 0)
Man. 23
BBRJKR6TbC$1 OAHi6IO 3 0 c H 0 B H H x B JI a c T H B 0 c T e ii Haii-
npocTimHX cl>iryp. u;io BJiaCTHBicTb ccl>OPMYJII06MO TaK:
VIII. Hiaiu 6u He 6ys rpuicynu.uc, icHye rpuicy'l'HUIC, UfO oopi8-
H'IOe UOXY 'fl aaiJaHOXY poaMiQfeHHi BiiJHOCHO 04HO'i nisnpR-MO'i.
11. IIAPAJIEJibHI IIPHMI
,lJ;Bi npBMi HR3HBRIOTbCS: napaJteJtb1£UMU, HKDW BOHH He nepeTH-
HRIOTbCB.
Ha MRJIIOHKY 24 noKaaaHo, BK aa AOnoMoroio KYTHHKa i JiiHiiiKH
npouecTH 11epea ARHY TO'IKY B npBMY b, napa11e111>Hy ARHiii nps-
Miii a.
Ilapa11e111>HicT1> npsMHX noaHa11a10T1> aHaKoM II. 3anHc a II b 'IH-
TaeT1>cs: •IlpsMa a napa11e111>Ha nps:Miii b•.
A
a
b
Man. 24 Man. 25

7 KJiac 16
0 c H 0 B H·a B JI a c TH Bi c Tb napaJieJibHHX np&MHX:
IX. l/epe3 To.,,xy, UfO ne iie31CUT'b "" iJamu np1U1iu, :.oxna
npo6ecTu na niioUfUKi ne 6iii'buie six oiJny npll.My, naJHJn.en.'l>NY
iJaniu.
r1 3 a A a q a (41). qH MO:llCe npS114a, ID;O nepeTHHae OAHY 3
V .u;~ox n~paneJibHHX np.RMHX, He nepeTHHaTH .u;pyry? IIo.acHiTb
Bl.ll;llOBl.n;b.
Po a B' .a a a H H s. Hexaif a i b - napaJieJibHi np.aMi i
Hexaif np.RMa c nepeTHHae np.aMy a B To11u;i A (MaJI. 25).
KoJIH 6 np.aMa c He nepeTHHaJia np.aMy b, TO 11epea TO'IKY A
npoxo,n;HJIH 6 ABi np&Mi, ID;O He nepeTHHRIOTb np.RMY b:
np.aMa a i np.aMa c. AJie aa BJiacTHBiCTIO napaJieJibHHX np.RMHX
u;e HeMOlKJIHBO. 0Tate, np.aMa c, .RKa nepeTHHae np.RMY a,
IlOBHHHa nepeTHHRTH i np.RMY b, napaJieJibHY lli:.
12. TEOPEMH I JJ;OBE)J;EHH.H
IlpaBHJILHiCTb TBep)l;ateHHSI npo BJlRCTHBiCTb Tl61 'IH iHmo'i
reoMeTpH'IHOI <PirypH BCTaHOBJII06TbCSI MipKyBaHH.RM. :u;e MipKy-
BaHH.R Ha3HBa6ThC.R aoeeaenHRM. A oaMe TBep,n;ateHH.R, .RKe AOB0-
,1.VITL, H83HBa6TbC.R reopeM.010. Haue,n;eMo npHKJI&,n;.
T e o p e M a 1.1. HXUfO npsi:M.a, sirw. Ne npoxoiJUT'b ...epe3 •oiJNy
3 Bepuiun TpuxyTKUKa, nepn'UlUU OiJNy 3 UotO CTOpiH, ro 801'4
nepeTuHae Tiii'l>xu oiJHy 3 iJBox inwux cropiH.
,lJ; o B e ,n; e H H .a. Hexa:H npBMa a He npoxo.a;HTL 11epea mo.a;Hy
a BepmHH TPHKYTHHKa ABC i nepeTHHae :Horo cropoHy AB (MB.JI. 26).
Ilp.aMa a poa6Huae nJIOID;HHY Ha ABi niBnJIOJD;HHH. To11KH A i B
JleataTL B pi3HHX niunJIOID;HHax, oc1dJILKH Bi.a;pi3oK AB nepeTHHae
npsMy a. To11Ka C Jle:IKHTL B o,n;Hi:H a D;HX niBIIJIOID;HH.
HKID;O TO'IKa C JiemHTb B o,n;Hiif nienJIOID;HHi a TO'IKOIO A, TO
ui.u;pi3oK AC He nepeTHHae npaMy a, a Bi.a;pi3oK BC nepeTHHae
u;ro npsMy (Man. 26, a).
HKID;O TO'IKa C nemHTL B o.a;Hi:H niunnoID;HHi a TO'IKOIO B, TO
ui.u;piaoK AC nepeTHHae np.aMy a, a ui.u;piaoK BC He nepeTHHae 'i'i
(MaJI. 26, 6).
B o6ox BHna,n;Kax np.aMa a nepeTHHae TiJihKH OAHH 3 ui.u;pi3KiB
AC a6o BC. Ou;e i yce AOBe.a;eHH.R.
<l>opMyJIIOBRHH.R TeopeMH 3BH'IRHHO CKJia,n;aeTLC.R 3 ABOX 11acTHH.
B o,n;Hi:H 11aCTHHi rouopHTLCB npo Te, ID;O .a;aHo. :u;s 11acTHHa Haau-

§ 1. 0cHOBHi BJiaCTHBOCTi Haii.npocTiWmt reoM0TpH'IHHX cl>iryp 17
B
a) 0)
Man. 26
BaGTbCJI yMoeo10 TeopeMH. B ,11;pyriii 11acTHHi roBopHTbCJI npo Te,
~o Ma6 6yTH ,ll;OBe,D;eHe. :u;a 11aCTHHa Ha3HBa6TbCJi 8UCH081'0M
TOOpeMH.
YMoBa TeopeMH 1.1 IIOJIHraG B TOMy, ~o npaMa He npoxo,D;HTb
11epea >KO,ll;HY 3 BepmHH TpHKYTHHKa i nepeTHHa6 O,ll;HY 3 iioro
CTOpiH. BHCHOBOK TeopeMH IIOJIHl'a6 B TOMy, ~o u;a npaMa nepeTH-
Ha6 TiJlbKH O,ll;HY 3 ,ll;BOX illmHX CTopiH TPHKYTHHKa.
13. AKCIOMH
Tsep,11;meHHR, RKi MiCTRTb cl>opMyJIIOBaHHR OCHOBHHX BJiaCTHBOC-
Teii HaiinpocTimHX «l>iryp, He ,ll;OBO,ll;RTbCSI i Ha3HBalOTbCH a1'CiOMa-
MU. CJIOBO aKcioMa IIOXO,ll;HTb Bi,11; rpeu;bKOrO CJIOBa a1'Cioc i 03H811a6
TBep,11;meHHSI, ~o He BHKJIHKas CYMHiBiB.
Ili,11; 11ac ,ll;OBe,D;eHHJI TeopeM ,11;03BOJISl6TbCR KOpHCTYB8THCJI
OCHOBHHMH BJI8CTHBOCTSIMH HaiinpocTimux «l>iryp, TOOTO aKCiOMa-
MH, a TaKom yme ,11;ose,11;eHHMH BJI8CTHBOCTSIMH, To6ro TeopeMaMH.
HisKHMH illmHMH anacTHBOCTSIMH «l>iryp, HaBiTb HK~o BOHH HaM
ap;aIOTbCSI O'leBH,ll;HHMH, KOPHCTyBaTHCb He MOlKHa.
IlpH ,ll;OBe,D;eHHi TeOpeM MO>KHa KOpHCTYB8THCSI MaJIIOHKOM, SIK
reoMeTpH'IHHM aanHCOM Toro, ~o BHpamasTLcs: c-1;1osaMH. He
A03BOJIH6TbCSI BHKOPHCTOByaaTH ni,11; 11ac MiPKYB8Hh ·~JiaCTHBOCTi
«l>iryp, SIKi BH,ll;HO 3 MaJIIOHKa, .ltK~O He MO>KHa o6rpyRTyBaTH 'ix,
CIIHp8IO'IHCb Ha aKCiOMH i reopeMH, ,ll;OBe,11;eui paHime.
y reoMeTpi'i nopa,11; 3 TaKHMH CJIOB8MH, RK aKcioMa i TeopeMa,
BlKHBaeTbCSI TaKOm CJIOBO •03H8'1eHHSI•. ,lJ;aTH 03Ha'teHHSI. qoro-
He6yp;h 03Ha11ae IIORCHHTH, ~o u;e TaKe.
'HanpHKJia,11;, rosopn&: •AaiiTe oaHa'leHHa TPHKYTHHKa•. Bi,11;-
noBi,D;aIOTh: •TpHKyTHHKOM Ha3HB86ThCSI cl>irypa, RKa CKJI8,11;86TbCSI

7 KJiac 18
3 TpbOX TO'IOK, ~o He Jie:>KaTb Ha op;Hiii npaMiii, i TpbOX Bip;piaKiB,
SIKi nonapHO CIIOJIY'IRIOTb ~ TO'IKH•.
,l.J;pyrHii npHKJiap;: •,l.J;aiiTe 03Ha'leHHSI napaJieJibHHX npSIMHX•.
Bi.zi;nosiAaIOTL: dlpaMi HaaHBaIOTbCSI napaJieJibHHMH, .HK~o BOHH
He nepeTHHaIOTLca•. BH Bme aHasTe oaHa11eHHS1 piBHOCTi Bi.zi;piaKiB,
piBHOCTi KYTiB i TPHKYTHHKiB.
? KOHTPO~hHI3AilHTAHH.H
•1. HaBep;iTL npHKJiap;H reoMeTpH'IHHX cpiryp.
2. HaaBiTL ocHOBHi reoMeTpH'IHi cpirypH Ha IIJIO~HHi.
3. HK rroaHa11aIOTLCS1 TO'IKH i rrpaMi?
4. CcpopMyJIIOHTe OCHOBHi BJiaCTHBOCTi HaJie:>KHOCTi TO'IOK i
IIpSIMHX.
5. IloacHiTb, ~o TaKe Bip;piaoK 3 KiHIJ;SIMH y p;aHHX TO'IKax.
6. CcpopMyJIIOHTe OCHOBHY BJiaCTHBiCTb poaMi~eHHa TO'IOK Ha
rrpaMiii.
7. CcpopMyJIIOHTe OCHOBHi BJiaCTHBOCTi BHMipIOBaHHSI Bip;piaKiB.
8. IIJ,o Ha3HBa6TbCSI Bi,zi;CTaHHIO Mi:>K ABOMa p;aHHMH TO'IKSMH?
9. HKi BJiacTHBOCTi Mas poa6HTTa IIJIO~HHH Ha p;Bi rriBIIJIO~HHH?
10. CcpopMyJIIOiiTe OCHOBHY BJiacTHeicTb poaMi~eHHSI TO'IOK BiA-
HOCHO IlpSIMOl Ha IIJIO~HHi.
11. IIJ,o TaKe rriBrrpaMa a6o rrpoMiHL? HKi niBrrpaMi liaaHBaIOTLca
,ll;OIIOBHSIJibHHMH?
12. HK 11oaHa11aIOTbCS1 rriBrrpaMi?
13. HKa cpirypa HaaHBaSTLca KYTOM?
14. HK 11oaHa11asTLca KyT?
15. HKHH KYT HaaHBaSTbC.H poaropHyTHM?
16. IloSICHiTb, ~o 03H8'186 BHpaa: tlliBrrpaMa rrpoxo,n;HTb Mi:>K CTO-
pOHSMH KyTa •.
17. B SIKHX OAHHHIJ,SIX BHMipIOIOTb KYTH i aa AOIIOMOrOIO SIKOro
iHCTpyMeHTa? IIoacHiTb, .HK BHKOHYIOTb BHMipIOBSHH.H.
18. CcpopMyJIIOiiTe ocHOBHi BJiaCTHBOCTi BHMip10BaHHa KyTiB.
19. CcpopMyJIIOHTe ocHoBHi BJISCTHBOCTi ui,n;KJiap;aHH.H ui.zi;piaKiB i
KYTIB.
20. IIJ,o TaKe TPHKYTHHK?
21. IIJ,o TaKe KYT TpHKYTHHKa rrpH ,n;aHiH BepurnHi?
22. HKi Bi,n;piaKH Ha3HB8IOTbCSI piBHHMH?

§ 1. OcHOBHi BJiaCTHBOCTi HaHnpocTiurnx reoMeTPH'IHHX «Iiiryp 19
23. HKi KYTH HaaHBaIOTLC.11 piBHHMH?
24. HKi TPHKYTHHKH HaaHBaIOTLc.11 piBHHMH?
25. HK noaHa11aIOTL Ha MaJIIOHKY Bi,11;n0Bi,11;Hi cTopoHH i KYTH piBHHX
TpHKYTHHKiB?
26. Ilo.11cHiTL aa MaJIIOHKOM 23 icHyBaHH.11 TPHKYTHHKa, ru;o ,11;opiB-
HI06 ,D;aHOMy.
27. HKi np.11Mi HaaHBaIOTLC.11 rrapaJieJILHHMH? HKHM aHaKoM rroaHa-
11aroTL rrapaJieJILHicTL rrp.11MHx?
28. Cq>0pMyJiroiiTe ocHOBHY BJiacTHBiCTL rrapaJieJILHHX rrp.llMHX.
29. HaBe,11;iTL rrpHKJia,11; TeopeMH•
• 3AAAq11
1. 1) IlpoBe,11;iTL np.11My. IloaHa11Te .11Ky-He6y,11;L T0111$Y A, ru;o Jie-
lKHTL Ha rrp.11Miii, i TO'IKY B, ru;o He JielKHTL Ha rrp.11Miii.
2) IlpoBe,11;iTL p;Bi rrp.11Mi a i b, ru;o rrepeTHHaIOTLC.11. IIoaHa11Te
TO'IKY C rrepeTHHY rrp.11MHx; TO'IKY A Ha rrp.11Miii a, ru;o He
JielKHTL Ha rrp.11Miii b; TO'IKY D, ru;o He JielKHTL Ha mop;Hiii
3 rrp.llMHX a i b.
2. IloaHa11Te Ha apKywi rrarrepy ,D;Bi TO'IKH. IlpoBep;iTL 11epea HHX
rrp.11My Bi.A pyKH. 3a p;on0Moro10 JiiHiiiKH rrepeBipTe rrpaBHJIL-
HiCTL rro6yp;OBH.
3. llH MOlKYTL p;Bi rrp.11Mi MaTH p;Bi TO'IKH rrepeTHHy? IIo.11cHiTL
BiAIIOBi,ll;L.
4. ,Il;JI.11 nepeBipKH rrpaBHJILHOCTi JiiHiiiKH BHKOpHCTOBYIOTL TaKHH
crroci6. l!epea ,D;Bi TO'IKH aa AOIIOMOroIO JiiHiiiKH rrpOB0,11;.llTL
JiiHiIO (MaJI. 27). IloTiM JiiHiHKY nepeBepTaIOTI, i 11epea Ti caMi
TO'IKH aHoBy rrpoBOA.llTL JiiHiro. HKru;o JiiHil He a6irJIHc.11, TO Jii-
HiiiKa HerrpaBHJILHa. Ha .llKiii BJiaCTHBOCTi rrp.11MHX rPYHTY6'l'LCH
TaKHii crroci6 nepeBipKH npaBHJIL-
HOCTi JiiHiiiKH?
5. IlpoBep;iTL rrp.11My a. IloaHa11Te Ha
np.11Miii ,n;Bi ,n;oBiJILHi TO'IKH A i B.
IloaHa11Te Terrep TO'IKY C TaK, ru;o6
TO'IKa A JiemaJia Mim TO'IKaMH Bi C.
~~~~~~~~~
L __','_1
Man. 27
1
B8r8TO 3~311 ~LOro n~py11HHK8 B3.RTO i3 WKiJibHHX ni~py'IHHKiB
i 3~8'1HHKiB MHHYJIHX poKiB, 30KpeM8 3 .reoMeTpih A. n. KuceJibOB8
i •36ipHHK8 3~8'1 3 reoMeTpih M. 0. Pn6KiH8.

7 IUIRC 20
6. IIpose,n;iTh npRMy a. IIoaHaqre Ha Hi:a ,n;Bi .n;osiJibHi ToqKH A i
B. IIoaHa'!Te Tenep J1Ky-He6y,n;L TO'IKY C Bi,n;pi3Ka AB.
7. ToqKa M nemHTb Ha npaMiii CD Mim TO'IKaMH Ci D. 3Haii,n;iTb
,n;oBmHHy Bi,n;pi3Ka CD, HKm;o: 1) CM= 2,5 CM, MD= 3,5 cM;
2) CM= 3,1 ,n;M, MD = 4,6 ,n;M; 3) CM= 12,3 M, MD =
= 5,8M.
8. IIoaHaqTe Ha npRMiii ,n;Bi ToqKH Ta Ha oKo cepe,n;HHY Bi.n;piaKa,
m;o cnonyqae ~ ToqKH. IIepeBipre npaBHJibHiCTb no6y,n;oBH BH-
MipIOBaHHJI aa ,ll;OilOMOrOIO JiiHiHKH.
9. TpH TO'IKH A, B, C nemaTb Ha o,n;Hiii npaMiii. Bi.n;oMo, m;o AB =
= 4,3 CM, AC = 7,5 CM, BC = 3,2 CM. qH Mome TO'!Ka A
nemaTH Mim TO'IKaMH Bi C? qH Mome TO'IKa C nemaTH Mim
TO'IKaMH Ai B? SIRa a Tphox TO'IOK A, B, C nemHTb Mim ,n;BOMa
iHIBHMH?
10. ToqKH A, B, C nemaTb Ha o.n;Hiii npaMiii. qH uanemHTb ToqRa B
Bi,n;pi3Ky AC, JIRm;o: AC= 5 CM, BC= 7 cM? IIoacHiTb Bi,n;-
nosi,n;L.
11. To'IKH A, B, C nemaTb Ha o,n;Hiii npaMiH. qH Mome TO'!Ka B poa-
,n;iJIJITH ToqKH Ai C, J1Km;o: AC= 7 CM, BC= 7,6 cM? Bi.n;no-
si.n;b IlOJICHiTb.
12. qH MomyTh TO'IKH A, B, C nemaTH Ha o,n;Hiii npaMiii., JIRm;o
AB = 1,8 M, AC = 1,3 M, BC = 3 M? IIoacHiTb Bi,n;noBi,n;b.
13. qH MomyTh TPH TO'IKH A, B, C nemaTH Ha o.n;Hiii npaMiii, JIRm;o
.n;osmHHa 6iJibmoro Bi.n;piaRa AB MeHma aa cyMy ,n;oBmHH Bi,n;-
piaRiB AC i BC? IIoacHiTh Bi.n;noBi,n;b.
14. To'IKH A, B, C nemaTb Ha o,n;Hiii npaMiii. 3Haii,n;iTb .n;osmHHY
Bi,n;pi3Ka BC, aKm;o AB=2,7 M, AC= 3,2 M. CKiJibKH poa-
B'aaRiB Mae aa,n;aqa?
15. Ha Bi,n;piaKy AB ,n;oBmHHOIO 15 M B3JITO TO'IKY C. 3Haii.n;iTh ,n;oB-
mHHH Bi,n;piaRisAC i BC, aRm;o: 1) Bi.n;piaoK AC Ha 3M .n;osmHii
aa Bi,n;piaoR BC; 2) Bi,n;piaoK AC y ,n;Ba paaH .n;osmHii aa Bi,n;piaoK
BC; 3) TO'IKQ. C - cepe,n;HHa Bi.n;piaKa AB; 4) ,n;oBmHHH Bi.n;pia-
RiB Ac i BC Bi,D;HOCJITbCJI, JIK 2 : 3.
16. IIpose,n;iTb npaMy i nooHaqTe HKy-He6y,n;b TO'IKY A, m;o He
nemHTb .Ha uiii npaMiii. IlooHaqTe Tenep ,n;Bi TOqKH Bi C TaR,
m;o6 Bi,n;piaoK AB nepeTHHaB npaMy, a Bi,n;piaoK BC He nepe-
THHRB ii.
17. ,Il;aHo npaMy i Tpu TO'IKH A, B, C, m;o He nemaTb Ha :u;i:H npaMiH.
Bi.n;oMo, m;o Bi,n;piaoK AB nepeTHHae npaMy, a Bi.n;piaoK AC He

§ 1. OcuoaHi BJIBCTHBOCTi uaiinpocTiWHX reoMeTPH'IHHX <Piryp 21
nepeTHHa6 ii. qn nepeTnHa6 npsMy BmpiaoK BC? BiAnOBi,u;L
noscHiTL.
18. ,IJ;aHo npaMy i 'IOTHPH TO'IKH A, B, C i D, m,o He nemaTL Ha o;iii
npSll'tliii. ~ nepeTHHa6 BmpiaoK AD npaMy, SIKm,o: 1) Bi,u;piaKu
AB, BC i CD nepe-rnHaIOTL npaMy; 2) BmpiaKH AC i BC nepe-
THHaIOTL npSIMy, a BmpiaoK BD He nepe-rnHa6; 3) BmpiaKH AB
i CD nepeTHHaIOTL npRMy, a BiApiaoK BC He nepe-rHHa6; 4) Bm-
piaKH AB i CD He nepeTHHaIOTL npsMy, a Bi,u;piaoK BC nepeTH-
Has; 5) BmpiaKH AB, BC, CD He nepeTHHaIOTL npsMy;
6) Bi,u;piaKH AC, BC i BD nepeTHHaIOTL npSIMy? IIoscHiTL
BmnoBiAL.
19. ,lJ;aHO n'SITb TO'IOK i npaMa, m,o He npOXO,!J;HTb qepea >KOAHY 3
o;HX TO'IOK. BiAOMO, m,o TPH TO'IKH nemaTL B OAHiH niBnnom,uHi
BiAHOCHO o;i6l npSIMOl, a ABi TO'IKH - B iHWiii niBilJIOID,HHi.
KomHy napy TO'IOK CilOJiy'leHO BmpiaKOM. CKiJILKH BmpiaKiB
nepe-rHHa6 npSIMy? BmnoBmL noacHiTL.
20. ,IJ;aHo npsMy a i TO'IKH A, X, Y, Z Ha o;iii npRMi:H (Man. 11).
BmoMO, ID;O TO'IKH xi y JiemaTb no O,!J;HH 6iK Bm TO'IKH A,
TO'IKH Xi Z TaKom nemaTL no OAHH 6iK Bi,u; TO'IKH A. HK poaMi-
1.D;eHi TO'IKH Y i Z Bi,!J;HOCHO TO'IKH A : no OAHH 6iK 'IH no piaHi
6oKn? IIoscHiTL BmnoBi,u;L.
21. IIoaHa'ITe ,u;Bi TO'IKH A i B. IIpoBe,u;iTL niBnpsMy AB.
22. Ha Bi,u;piaKy AB B3SITO TO'IKY C. Cepe,u; niBnpSIMHX AB, AC, CA,
CB HaaBiTL napy niBnpsMHX, aKi a6iraIOTLCSI, i napy ,u;onoB-
HHJibHHX niBnpaMnx. IIOSicHiTL BmnoBmL.
23. IIpoBe,u;iTL a o,u;His'i TO'IKH Tpu AOBiJILHi npoMeHi. BnaHa'ITe Ha
OKO KYTH, YTBopeHi D;HMH npoMeHaMH. IIepeBipTe Barui Bi,u;no-
Bi,u;i, BHMipIOIO'IH KYTH TpaHcnopTHpOM. BnKoHaiiTe BnpaBy
IlOBTOpHO.
24. IIpoMiHL a npoXOAHTL Mim cTopoHaMH KyTa (ed). 3HaHAiTL
KYT (ed), SIKID;O: 1) L(ae) = 35°, L (ad)= 75°; 2) L (ae) =
= 57°, L(ad)= 62°; 3) L(ae)= 94°, L(ad)= 85°.
25. qH Mome npoMiHL e npoXOAHTH Mim CTOpOH8MH KyTa (ab),
aKm,o: 1) L (ae) = 30°, L (eb) = 80°, L (ab)= 50°;
2) L (ae) = 100°, L (eb) = 90°; 3) KYT (ac) 6inLmnii aa
KYT (ab)?
26. Mim CTOpoHaMH KYTa (ab), m,o ,u;opiBHIOG 60°, npoxo,u;HTL
npoMiHL e. 3Haii,u;iTL KYTH (ae) i (be), SIKID,O: 1) KYT (ae) Ha 30°
6iJILWHH aa KYT (be); 2) KYT (ae) B ABa pa3H 6iJibWHH 38 KYT

7 KJiac 22
(be); 3) npoMiHL c ,D;iJIHTL KYT (ab) IIOIIOJiaM; 4) rpap;ycHi MipH
KYTiB (ae) i (be) Bi,ll;HOCSITLCSI, SIK 2 : 3.
27. IIpoBe,n;iTL 11pS1My. 1Io:ma11Te Ha ~ npS1MiH S1Ky-He6y,n;L TO'IKY
A. IIoTiM Bi3LMiTL Ha OKO TO'IKY B Itie'i 11pS1Mo'i TaK, ~06 AB =
= 5 CM. IIepeBipTe TO'IHiCTL 1106y,n;OBH TO'IKH B JiiHiHKoIO.
BuKoHaiiTe B11paBy noBTOpHo, SIK~o: 1) AB= 3 CM; 2) AB=
= 7 CM; 3) AB= 10 CM.
28. IIo6y,u;yiiTe Ha OKO KYTH 30°' 45°, 60°' 90°. IIepeBipTe TO'IHiCTL
1106y,n;OBH TpaHCIIOpTHpOM. BHKOHaiiTe BllpaBy IIOBTOPHO.
29. l:Ju icHya Ha 11iB11pS1MiH AB TaKa T011Ka X, Bi,u;MiHHa Bi,n; B, ~o
AX = AB? IIOSICIDTL Bi,ll;IIOBi,ll;L.
30. Ha 11poMeHi AB Bi,n;KJiap;eHo Bi,n;piaoK AC, MeHIIIHH aa Bi.u;piaoK
AB. HKa a TpLox TO'IOK A, B, C JieJKHTL Mim ABOMa iHIIIHMH?
Bi,u;110Bi,ll;L IlOSICHiTL.
31. Ha 11poMeHi AB 11oaHa11eHo TO'IKY C. 3Haii,n;iTL ,n;oBJKHHY
Bi,n;piaKa BC, SIK~o: 1) AB = 1,5 M, AC= 0,3 M; 2) AB=
= 2 CM, AC = 4,4 CM.
32. IIo6y,u;yiiTe Ha oKo TPHKYTHHK a piBHHMH cropoHaMH (piBHo-
CTOpoHHiH TPHKYTHHK). IIepeBipTe To11HicTL 1106yp;OBH BHMip10-
BaHHS1M CTOpiH.
33. Ha cTopoHi AB TPHKYTHHKa ABC B3SITO TO'IKY D. l:JoMy .u;o-
piBHIOa CTOpoHa AB TpHKYTHHKa, SIK~o AD = 5 CM, a BD =
= 6cM?
34. Ha cTopoHi AB TpHKYTHHKa ABC B3SITO TO'IKY D. l:JoMy .u;opiB-
HIOa KYT c TPHKYTHHKa, SIK~O L ACD = 30°, a L BCD= 70°?
35. HaKpecJiiTL ,n;oBiJILHHH TpHKYTHHK. IIo6y,u;yiiTe BiA PYKH, Ha
oKo, TPHKYTHHK, ~o p;opiBHI06 iioMy. IIepeBipTe npaBHJILHiCTL
1106y,n;OBH, BHMipIOIO'IH Bi,ll;IIOBi,n;Hi KYTH i CTOpoHH. BHKOHaHTe
BllpaBy IIOBTOpHO.
36. TpHKYTHHKH ABC i PQR piBHi. Bi,u;oMo, ~o AB = 5 cM, BC =
= 6 CM, AC = 7 CM. 3Haiip;iTL cTopoHH TPHKYTHHKa PQR.
IIOSICHiTL Bi,ll;IIOBi,ll;L.
37. TpHKYTHHKH ABC i PQR piBHi. KyTH .u;pyroro TpHKYTHHKa
Bip;oMi: LP= 40°, L Q = 60°, LR= 80°. 3Haiip;iTL KYTH
TPHKYTHHKa ABC.
38. TpuKyTHHKH ABC i PQR piBHi. Bi.u;oMo, ~o cTopoHa AB .u;opiB-
HI06 10 M, a KYT C = 90°. qoMy ,u;opiBHI06 cTopoHa PQ i KYT R?
IIOSICHiTL Bi,u;IIOBi,u;L•.
39. TpHKYTHHKH ABC, PQR i XYZ piBHi. Bi,o;oMo, ~o AB = 5 cM,

§ 1. OcHOBHi BJiaCTHBOCTi HaiirrpocTiurnx reoMeTpH'lHHX <l>iryp 23
QR = 6 CM, zx = 7 CM. 3Haii,u;iTb pemTy CTOpiH Km1rnoro TpH-
KYTHHKa.
40. ,D;aHo TpHKYTHHK ABC. qH icHye iHmHii TPHKYTHHK ABD, w;o
AOpiBHIOe AaHoMy?
41. 'IH Mome rrpSIMa, w;o rrepeTHHa6 OAHY a ABOX rrapaJieJibHHX
IIPSIMHX, He rrepeTHHaTH ,u;pyry? IlOSICHiTb Bi,u;rroBiAb.
42. ,lJ;aHO ABi IIPSIMi, w;o rrepeTHHaIOTbCSI. 'IH MOmHa rrpoBeCTH Tpe-
TIO rrpSIMy, rrapaJieJibHY KOmHiH a ABOX AaHHX?
43. 'IH Mome rrpSIMa, w;o He rrpoxo,n;HTb 11epea moAHY a BepmHH TpH-
KYTHHKa, rrepeTHHaTH KOmHy :Horo CTOPOHy? qoMy?
44!' 1
,D;aHo 'IOTHPH TO'IKH: A, B, Ci D. BiAoMo, w;o TO'IKH A, B, C
JiemaTb Ha OAHiii rrpSIMiH i TO'IKH B, C, D TaKom JiemaTb Ha
o,u;Hi:H IIPSIMiH. ,ll;oBe,n;iTb, w;o BCi 'IOTHPH TO'IKH JiemaTb Ha
o,u;HiH rrpSIMiH.
45!' ,lJ;aHo 'IOTHPH rrpSIMi a, b, c i d. Bi,u;oMo, w;o rrpSIMi a, b, c rrepe-
THHaIOTbC.fl B o,u;HiH TO'IIV ii rrpSIMi b, c, d TaKom rrepeTHHa-
IOTbCSI B o,u;HiH TO'IIV. ,ll;oBe,u;iTb, w;o BCi 'IOTHpH rrpSIMi rrpo-
XOASITb qepea OAHY TO''IKY.
46!' To11KH A, B, C, D He JiemaTb Ha o,u;HiH rrpSIMi:H. Bi,u;oMo, w;o
rrpSIMa AB rrepeTHHa6 Bi,u;piaoK CD, a rrpSIMa CD rrepeTHHa6
BiApiaoK AB. ,ll;oBe,n;iTb, w;o Bi,u;piaKH AB i CD rrepeTHHaIO-
TbCSI.
47!' ,D;aHo TpHKYTHHK ABC. Ha cTopoHi AC BaSITo TO'IKY B1, a Ha
cTopoHi BC - TO'IKY A1. ,lJ;oBe,u;iTb, w;o BiApiaKH AA1 i BB1
rrepeTHHaIOTbCSI (MaJI. 28).
48!' Bi,u;piaKH AB i CD He JiemaTb Ha o,u;HiH rrpSIMiH i rrepeTHHaIO-
[
B A D B 0 8 [
Man. 28 Man. 29 Man. 30
I 3ipO'IKOIO (*) 003Ha'leHO 38A8'1i niAsm:qeHOl CKJI8AHOCTi.

7 KJiac 24
TbC.H B To11~i E. ,D.one,n;iT&, m;o ni,n;piaoK AC He nepeTHHae
np.HMY BD (Ma.JI. 29).
49!' ,D.one,n;iTb, m;o KOJIH npoMiHb, JIKHH BHXO,ll;HTL 3 BepII!HHH
KyTa, nepeTHHae ni,n;piaoK AB a KiH~MH Ha cropoHax KyTa,
TO BiH nepeTHHae: 1) ni,n;piaoK AC a KiH~MH Ha cropoHax
KyTa; 2) ,n;onin&HHH ni.zi;piaoK CD a KiH~BMH Ha cTopoHax
KyTa (Man. 30).
50. ,D.one,n;iT&, m;o ,n;ni np.HMi a6o napanenLHi, a6o nepeTHHaIOT&cs
B O,ll;Hiii TO'I~.
51!' To11KH A i C Ha.JiemaT& np.HMi:H a. Ha niBnps:Miii CA ni,n;Kna-
,n;eHo ni,n;piaoK CB, 6in&mHii aa niApiaoK CA. 1) HKa a Tp&ox
TO'IOK A, B, C nemHT& Mim ,n;BoMa iHmHMH? IIoscHiT& niAno-
BiA&. 2) ,D.one.zi;iT&, m;o TO'IKa A poa6Hnae nps:My a Ha JlBi
ninnp.HMi: AB i AC.
§ 2. CYMIJKHI I BEPTHKAJibHI KYTH
14. CYMDKHI KYTH
0 3 Ha 'I e H H .fl. ,lJ.Ba KYTH Ha3HB8IOTbC.H cyMi:Jtt:HUMU, JIKIIl;O
B HHX O,ll;H8 CTOpOHa cnin&Ha, a iHmi CTOPOHH ~HX KYTiB e ,ll;OilOB-
HaJibHHMH niBnp.HMHMH.
Ha ManIOHKY 31 KYTH (a1b) i (a2b) cyMimHi. y HHX CTOpOHa b
cnin&Ha, a cTopoHH a1 i a2 e ,n;onoBH.Hn&HHMH ninnps:MHMH.
Hexaii C - TO'IKa Ha np.HMiH AB, m;o nemHT& Mi:>K TO'IKaMH A i
B, a D - T011Ka, m;o He nemHTb Ha np.HMiii AB (Man. 32). To,n;i KYTH
BCD i ACD cyMimHi. Y HHX cTopoHa CD cninLHa. CTopoHH CA i
CB - ,n;onoBHBn&Hi niBnp.HMi nps:Moi: AB, ocKiJI&KH To1uw A i B
~HX niBnp.HMHX poa,n;ineHi IlO'laTKOBOIO TO'IKOIO c.
b
A 8
A a, [
Man. 31 Man. 32

§ 2. CyMiJKHi i sepTHK8JlbHi KYTH 25
Teo p e M a 2.1. CyMa cyMi.31cHux JCyTi6 iJopisff.K>e 180°.
Ao e e 11. e H H H. Hexaii L (a1b) i L(a2b) - 11.aHi cyMimHi
KYTH (MaJI. 31). IlpoMillL b npoXOJJ.HTb Mim CTOpoHaMH a1 i a2 poa-
ropHyToro KyTa. ToMy CYMa KyTiB (a1b) i (a2b) 11.opiBHIOO poaropHy-
TOMY KyTy, T06To 180°. TeopeMy 11.oee11.eHo.
fa TeopeMH 2.1 BHilJIHBae, IIJ,O 1COJ£U 06a JCYTU pi6Hi, TO cyMi:HCHi
3 ff.UMU JCYTU TalCO:llC pi6ff.i.
fa TeopeMH 2.1 TaKOJK BHilJIHBae, IIJ,O 1COJ£U lCYT 1U! po3ZOPHYTUU,
TO UoZO zpaiJycff.a Mipa MeHfll,4 6W 180°.
/'f 3 a 11. a 'I a (3). 3Haii11.iTb cyMiJKHi KYTH, HKIIJ,O OJJ.HH 3 HHX
'LJ Y 11.ea paaH 6iJILmHii aa 11.pyrHii.
Po a e' g a a H Ha. IloaHa'IHMO rpaJJ.ycHy Mipy MeHmoro
KyTa 11epea x. To,zti rpaJJ.ycHa Mipa 6i.n:bmoro KyTa 6y11.e 2x.
CyMa Kyris 11.opieHioe 180°. Orme, x + 2x = 180°, 3x =
= 180°. 3BiJJ.CH x · 60°. BHXOJJ.HTb, Hami CYMiJKHi KYTH 11.0-
piBHIOIOTb 60° i 120°.
KyT, w,o 11.opieH10e 90°, HaaHeaeTbCH npR.MuM KyroM.
3 TeopeMH npo CYMY CYMiJKHHX KyTiB BHilJIHB&G, IIJ,O lCYT' cy.111i:11e-
HU'4 3 nplUIUM lCYTOM, e npR.Mu4 JCyT.
KyT, MeHmHii aa 90°, HaaHeaeTbC.R zocrpuM JCyroM. KyT, 6iJIL-
mHii aa 90° i MeHmHii eiJJ. 180°, HaausaeTbC.R rynuM.
Lrocmpuu KIJm
MaJI. 33
OcKiJibKH cyMa cyMiJKHHX KyTiB 11.opiBHIOe 180°' TO KyT, cyMiJK-
HHH a rocTPHM KyToM, TynHii, a KyT, cyMimmm a TynuM,- rocT-
pHii. Ha M&JIIOHKY 33 ao6pameHo TPH BHJJ.H KyTiB.
15. BEPTBKAJILHI KYTH
0 a Ha 'I e.H Ha. Aea KYTH HaaHBaIOTbC.R BepruKaAbH.UMU,
.RKIIJ,O CTOPOHH OJJ.HOrO KyTa e lJ.OilOBH.RJibHHMH nienpaMHMH CTOpiH
11.pyroro.

7 KJiac 26
~h,
MaJI. 34 MaJI. 35
Ha MaJIIOHKY 34 KyTH (a1b1) i (a2b2) BepTHKaJibHi. CTopoHH a2
i b2 APYroro KyTa - AOilOBHHJibHi niBnpHMi cTopiH a, i b, nepmoro
KyTa.
Teo p e Ma 2.2. Beyru1"J.J£'bH.i xyru piBH.i.
,IJ; o Be A e H H H. Hexa:H (a1 b1) i (a2b2) - AaHi BeprHKaJibHi KYTH
(MaJI. 34). KyT (a1b2) cyMhKHHH 3 KYTOM (a1b1) i KYTOM (a2b2). 3Bi,lCH
aa TeopeMOIO npo CYMY cyMi)l(HHX KYTi.B po6HMO BHCHOBOK, ~o KO)l(-
HHH a KyTiB (a1b1) i (a2b2) AOllOBHIOO KYT (a1b2) AO 180°, To6To KYTH
(a1b1) i (a2b2) piBHi. TeopeMy AOBeAeHo.
3 a A a 'I a (9). CyMa ABOX KyTiB, YTBOpeHHx y peayJILTaTi
nepeTHHY ABOX npHMHX, AOPiBHIOe 50°. 3HaHAiTL ~i KYTH.
p 0 3 B' H 3 a H H s:. ,IJ;Ba KYTH, HKi YTBOpHJIHCH B peayJILTa-
Ti nepeTHHY )BOX nps:MHX, a6o cyMi)l(Hi, a6o BePTHK8JILHi
(MaJI. 35). ,IJ;aHi KYTH He MO)l(YTb 6yTH cyMhKHHMH, OCKiJibKH
'ix cyMa ,lOpiBHIOe 50°' a cyMa cyMi)l(HHX KYTiB CTaHOBHTb
180°. OT)l(e, BOHH BepTHKaJibHi. OcKiJii.KH BepTHKaJibHi KYTH
piBHi ii aa YMOBOIO lX cyMa 50°' TO KO)l(HHH 3 HHX AOPiBHI06
25°.
16. IIEPIIEllNfKYJUIPHI IIP.JIMI
Hexaii a i b - ABi npHMi, ~o nepeTHHaIOTbCS: y TO'I~ A
(MaJI. 36). KomHa a ~Hx npHMHx AiJIHTbcs TO'IKOIO A Ha ABi niB-
a
MaJI. 36
npaMi. IIiBnpaMi OAHisi'. npaMoi'. YTBo-
pIOIOTb 3 niBnpSIMHMH )pyroi'. npHMOl
'IOTHPH KyTH. Hexa:H a - OAHH a ~x
KyTiB. BYAL-SIKHH 3 pellITH TpbOX KyTiB
6y)e a6o cyMimHHM 3 KYTOM a, a6o
BepTHKaJlbHHM 3 KYTOM a. 3BiACH BH-
llJIHB86, ~o KOJIH OAHH 3 KyTiB npSIMHH,
TO pemTa KYTiB Te)I( npaMi. y ~bOMY
BHllaAKY roBOpSITb, ~o npaMi nepeTHHa-
IOTbCH niA npSIMHM KYTOM.

§ 2. CyMhKHi i BepTHK&JILHi KYTH 27
b c,
A a a a,
MaJI. 37 MaJI. 38
0 a Haq e H Hg. ,IJ;si npgMi Ha3HBaIOTbcg nepneniJU1cyJLapnu-
Mu, $1K~O BOHH nepeTHHaIOTbC$1 ni,u, np$1MHM KYTOM (MaJI. 37).
IlepneH,11,HKyJigpHiCT& np$1MHX Il03Ha'laIOTh 3H8KOM _L. 3anHC
a _L b 'IHT86TbC$1 TaK: tllp$1Ma a nepneH,U,HKYJI$1pHa ,11,0 np$1MOl b••
Teo p e Ma 2.3. ·qe-pea K031CHY T01'KY npJUCoi M031CHa npoBecru
nepneH.iJUKYlf.R.PHY iJo H.ei npJUCy i iJo f'OZO 3IC Tilf.b"ICU oiJHy.
,n; o B e .u. e H H a. Hexa:H a - ,u,aHa npaMa i A - TO'IKa Ha Hi:H.
IloaHa'IHMO qepea a1 0,11,HY 3 niBnpHMHX np$1MOl a 3 Il0'18TKOM
y TO'lltj. A (MaJI. 38). Bi,11,KJia,11,eMo Bi,u, niBnpaMoi a1 KYT (a 1b1), ~o
,u,opiBHIOe 90°. To,u,i np$1Ma, gKa MiCTHTh npoMiHh bi, 6y,u,e nepneH-
AHKYJIHpHa ,u,o npaMoi a.
IlpHilYCTHMO, ~o icHye iHma npaMa, JIKa Te:>K npOX0,11,HTb qepea
TO'IKY Ai nepneH,U,HKyJiapua ,u,o npaMoi a. Iloaua11HMO 11epea c, nis-
npaMy Jtj.ei np$1MOi, ~o Jie:>KHTb B 0,11,Htii niBilJIO~HHi 3 npoMeHeM bI·
KYTH (a1b1) i (a1c1), Komuuii a $1KHX ,u,opiBHIOe 90°, Bi,u,KJia,11,eHo
B 0,11,HY niBnJIO~HHY Bi,11, niBnpSIMOl aI· A.rre Bi,11, niBnp$1MOl aI y ,u,aHy
A
B a
MaJI. 39 MaJI. 40

7 KJiac 28
niBilJIOil~llHY MOmHa BiAKJiaCTH TiJibKH OAHH KYT, mo AOPiBHI06 90°.
ToMy He Mome 6yTH iHmoi npsMoi, HKa npoxoAHTb 11epea TO'IKY A
i nepneHAHKYJIHpHa AO npHMoi a. TeopeMy AOBeAeHo.
0 a H a 11 e H H H. llepnenauH:yJuipoM. AO AaHoi npaMoi HaaHBas-
TbCa Bi.u;piaoK npaMoi, nepneHAHKYJI&pHoi AO AaHoi npaMoi, BKHii
Ma6 OAHHM ia CBOlX KiH~iB TO'IKY ix nepeTHHY. IJ;eii KiHe~b BiApiaKa
Ha3HBa6TbCJI OCH06010 nepneHAHKYJIHpa.
Ha MaJIIOHKY 39 nepneHAHKyJiap AB npoBe,u;eHo a TO'IKH A AO
npaMoi a. To11Ka B - ocHoBa nepneHAHKyJiapa. ,lJ;JIH no6yAOBH
nepneHAHKyJiapa KOpHCTYJOTbCJI KOCHH~eM (MaJI. 40).
17. )J;OBE)J.EHHH BI)J. CYIIPOTHBHOrO
Cnoci6 AOBeAeHHa, aKHii MH aaCTocyBaJIH AO TeopeMH 2.3, Ha-
3HBa6TbCH AOBeAeHHJIM BiA cynpOTHBHOro. IJ;eii cnoci6 AOBeAeHH.H
IlOJiaras B TOMy, mo Cil0'18TKY po6HMO npHnymeHH.H, npOTHJiemHe
TOMy, &Ke CTBepAmY6TbCH TOOpeMOIO. IIOTiM MiPKYB8HHHMH,
CllHpaIO'IHCb Ha aKciOMH i Bme AOBeAeHi TeopeMH, npHXOAHMO AO
BHCHOBKY, mo cynepe'IHTb a6o YMOBi TeOpeMH, a6o OAHiii 3 aKcioM,
a6o AOBeAeHiii paHiIIIe TOOpeMi. Ha ~iii OCHOBi po6HMO BHCHOBOK,
mo Harne npunymeuHa HenpaBHJibHe, a TOMY npaBHJibHe TBepAmeH-
Ha TeopeMH.
IIoacHHMO ~e Ha npHKJiaAi AOBeAeHHa TOOpeMH 2.3. Y TeopeMi
CTBepAmY6TbCH, mo qepea KOmHy TO'IKY npHMOi MOmHa npoBeCTH
TiJibKH OAHY nepneHAHKYJI&pHy AO Hei npaMy. IlpHnycTHBmH, mo
T8KHX np&MHX MOmHa npoBecTH ABi, MH AiHillJIH BHCHOBKY, mo BiA
A8Hoi niBnpaMoi y AaHy niBIIJIOmHHY MOmHa Bi,u;KJI8CTH ABa
KYTH a OAHis10 i Tis10 caMoIO rp&AYCHOIO Mipo10 (90°). A ~e cynepe-
'IHTb 8KCiOMi Bi,u;KJiaAaHHH KyTiB, aa JIKOIO Bi,u; AaHoi niBnp.HMOi
y AaHy niBDJIOmHHY MOmHa BiAKJI8CTH JIHWe OAHH KYT 3 AaHOIO
rp8AYCHOIO Mipo10.
18. BICEKTPHCA KYTA
0 a H a q e H H a
1
• BiceH:rpuco10 KYTa H83HB86TbC& npoMiHb,
JIKHH BHXOAHTb 3 BepnrnHH KYT8, npoXOAHTb Mim iioro CTOpOH8MH
i AiJIHTb KYT ll0IlOJI8M.
1 .D:aJii cnouo •oaHa'leHHH• aanHcyuaTH He 6y,u;eMo, a oaHa'lyBaHe
noH.HTT.11 BHAiJIHTHMeMo KypcueoM.

§ 2. CyMilKHi i BepTRKSJihHi KYTR
Ha MaJIIOHKY 41 BH 6a'IHTe KYT (ab).
IlpoMiHb c BHXO,ll;HTb 3 BepmHHH KyTa,
IIpoXO,ll;HTb Mim ii:oro cropoHaMH i ,zi;iJIHTb
KYT IIOilOJiaM:
L(ac)=L(bc)= L~ab).
3 a ,zi; a q a 17. ,ZJ;oee,zi;iTh, ~o 6i-
ceKTpHca KyTa YTBOpIOe 3 ii:oro CTOpo-
HaMH KYTH He 6i11bmi aa 90°.
Po a e' a a a H H a. HK MH aHae-
29
a
Man. 41
MO, rpa,zi;ycHa Mipa 6y,zi;h-aKoro KyTa He 6i11bma aa 180°. ToMy
rroJIOBHHa ii:oro He '6i1Ibma aa 90°.
19. ~O IIOTPIBHO POBHTH,
~OB )J;OBPE BCTHrATH 3 rEOMETPII
Ili,zi; qac BHB'leHua reoMeTpii HeaHaHHa qoro-ue6y,zi;h a npoii:,zi;euo-
ro MaTepia11y MOme CTaTH npH'IHHOIO uepoayMiHH$1 HOBOro MaTepi-
aJiy. HaBe,zi;eMo npHKJia,zi;.
IlpHIIYCTHMO, Ha ypo~i B'IHTeJib AOBO,ll;HTb TeopeMy npo piBHiCTb
eepTHKaJihHHX KyTiB. HK DH aHaeTe, ,zi;na ,zi;oBe,zi;eHHa ~iei TeopeMH
BHKOpHCTOBYIOTb 03Ha'leHH$1 cyMimHHX KYTiB i TeopeMy rrpo CYMY
cyMi)KHHX KyTiB. HK~o DH He aHaeTe, aKi KYTH Ha3HB8IOTbCa cy-
Mi)KHHMH, He 3HaeTe TeopeMH npo cyMy cyMi)KHHX KyTiB, TO BH He
apoayMiere ~oro ,zi;oBe,zi;eHHa. B peayJibTaTi ~eii: ypoK 6y,zi;e ,zi;na Bae
,zi;apeMHO BHTpaqe11HM qacoM. I ,zi;o Damoro ueaHauua npo CYMi)KHi
KYTH ,zi;o,zi;aeTbC$1 He3HaHH$1 TeopeMH npo piBHiCTb BepTHKRJibHHX
KYTiB.
ToMy, ~06 ,zi;o6pe BCTHraTH 3 reoMeTpii, Tpe6a 3HaTH OCHOBHi
peaynhTRTH BHB'leHoro MaTepiany. A ,zi;na ~oro cni,zi; noerop10BaTH
BHB'leHHH MaTepian aa KOHTPOJibHHMH 38IIHT8HHSMH.
IloBTOPIOBaTH rrpoii:,zi;eHHii MaTepiaJI 38 KOHTpoJibHHMH 38IIHTaH-
HRMH MomHa TaK. IIpo'IHTaii:Te aanHTRHHa. YcB~OMTe ii:oro aMiCT.
HK~o Tpe6a ,zi;aTH oaua'leHHa aKoi-He6y,zi;h cl>irypH, ,zi;aii:Te TaKe
oauaqeuHa npo ce6e.
RopHCHo apo6HTH B~ pyKH MaJIIOHOK cl>irypH, oaaa11eHHa
a:Koi MaeTe ,zi;aTH. SIK~o y aanHTRHHi ii:,zi;eTbca rrpo TeopeMy, ccl>op-
MYJIIOii:Te ii, a'acyiiTe, B 'IOMY nonarae YMOBa i BHCHOBOK Teo-
peMH.

7 KJiaC 30
3po6iTb BiA PYKH MRJIIOHOK, S:IKHH iJIIOCTpye 3MiCT TeopeMH• .D;oBe-
,Zl;eHHS:I TeopeMH npH KOJKHOMY OOBTOpeHHi po6HTH He o6oB'.H3KOBO.
IIoBTOPIOHTe npoii,Z1;eHHii MarepiaJI iu;opaay, KOJIH y npou;eci
BHB'leHHS:I HOBoro MaTepiaJiy BH BH.HBHJIH He3HRHH.H qoro-He6y,Zl;L.
? KOHTPOnbHI3AilHTAHHH
•1. .ffKi KYTH HR3HBRIOTbCS:I cyMhKHHMH?
2. .D;oBe,Zl;iTL, ID;O cyMa cyMimHHX KyTiB ,Zl;OpiBHI06 180°.
3. .D;oBe,Zl;iTb, ID;O KOJIH ,Zl;Ba KYTH piBHi, TO cyMimHi 3 HHMH KYTH
Tem piBHi.
4. HKHH KYT HaaHBaeTbCS:I npS:IMHM (rocTpHM, TynHM)?
5. ,lJ;oBe,ll;iTb, iu;o KYT, cYMhKHHii a npS:IMHM, npaMHii.
6. .ffKi KYTH HR3HBRIOTbCH BepTHKaJILHHMH?
7. ,lJ;oBe,Zl;iTL, iu;o BepTHKRJibHi KYTH piBHi.
8. ,lJ;oBe,Zl;iTb, ID;O KOJIH OPH nepeTHHi ,Zl;BOX npaMHX O,Zl;HH 3 KyTiB
npHMHH, TO pemTa KYTiB Tem npaMi.
9. .HKi npaMi HR3HBaIOTbCH nepneH,Zl;HKYJIS:lpHHMH? .HKHH 3HaK
BHKOpHCTOBY6TbCH ,Zl;JIH noaHa'leHHS:I nepneH,Zl;HKYJI.HPHOCTi
npS:IMHX?
10. ,lJ;oBe,Zl;iTb, ID;O qepea 6y,Zl;L-HKY TO'IKY npHMoi MO>KHa npoBeCTH
nepneH,Zl;HKYJIHPHY ,Zl;O Hei npaMy i TiJibKH 0,Zl;Hy.
11. ID;o TaKe nepneH,Zl;HKYJIHP ,Zl;O npS:IMoi?
12. Ilo.HCHiTb, y 'IOMY IlOJiarae ,ll;OBe,Zl;eHH.H BiA cynpoTHBHOro.
13. ID;o HR3HBR6TbCH 6iceKTpHCOIO KyTa?
1. 3HRH,Zl;iTb KYTH, cyMiJKHi 3 KYTRMH 30°; 45°; 60°; 90°.
2. 'tlH MomyTb ,Zl;Ba cyMi>KHi KYTH 6YTH OOH,Zl;Ba: 1) rocTpHMH;
2) TYOHMH; 3) npS:IMHMH? 06xtpyHTYHTe BiAnoBiAb.
3. 3HRH,Zl;iTb cyMimHi KYTH, HKID;O O,Zl;HH 3 HHX y ,Zl;Ba paaH 6iJiblliHH
Bi,ZI; ,Zl;pyroro.
4. 3Haii,Zl;iTb cyMimHi KYTH, S:IKID;O: 1) O,Zl;HH 3 HHX Ha 30° 6iJibWHH
Bi,ZI; ,Zl;pyroro; 2) ix piaHHD;.H p;opiBHIOO 40°; 3) O,Zl;HH 3 HHX y TPH
paaH MeHmHii Bi,ZI; ,Zl;pyroro; 4) BOHH piBHi.
5. .HKHii KYT yTBopIOIOTL rop;HHHa i XBHJIHHHa cTpiJIKH ro,Zl;HHHH-
Ka, KOJIH BOHH llOKR3YIOTb! 1) 6 l'O,ZI;; 2) 3 l'O,ZI;; 3) 4 ro,Z1;?

§ 2. CyMilKHi i BepTHKll.ll&Hi KYTH 31
6. 3Ha~iTb cyMhKHi KYTH, $1KUJ;O ix rpa,lyCHi MipH BiAHOC$1TbC$1,
$1K! 1) 2 : 3; 2) 3 : 7; 3) 11 : 25; 4) 22 : 23.
7. 0AHH 3 KyTiB, YTBOpeHHX B peayJibTaTi nepeTHHy ABOX np$1MHX,
AOpiBHIOe 30°. qoMy AOpiBHI06 pemTa KyTiB?
8. qOMY AOPiBHIOe KyT, $1KUJ;O ABa cyMi)l(Hi 3 HHM KYTH CTaHOB-
Jl$1Tb B CyMi 100°?
9. CyMa ABOX KYTiB, yTBopeHHX B peayJILTaTi nepeTHHY ABOX npa-
MHX, AOPiBHI06 50°. 3HaHAiTb ~ KYTH.
10. 0AHH 3 KyTiB, YTBOpeHHX B peayJibTaTi nepeTHHy ABOX np$1MHX,
y 4 pa3H 6iJibWHH BiA Apyroro. 3HaHAi'l'b ~ KYTH.
11. 0AHH 3 KyTiB, YTBOpeHHX B peayJibTaTi nepeTHHY ABOX npaMHX,
Ha 50° MeHmHH: siA APYroro. 3Ha~iTb ~ KYTH.
12. 3HaiiAiTb KYTH, HKi YTBOpIOIOThC.11 B peayJILTaTi nepeTHHY ABOX
npa:MHX, a:Kw;o cyMa TpLox a HHX AOPiBHI06 270°.
13. ,D;oseAiTL, w;o KOJIH TPH a 'IOTHpbox KYTiB, yTsopeHHx B peayJIL-
TaTi nepeTHHY ABOX np.llMHX, piBHi, TO npaMi nepneffAHKyJia:pHi.
14. SIK JiiHiHKOIO nepesipHTH, 'IH npaMHH KYT y KOCHH~a (MaJI. 42)?
15. qoMy AOPiBHIOe KYT Mi)!( 6iceKTPHCOIO i cropoHOIO AaHoro KyTa,
HKHH ,lOpiBHI06: 1) 30°; 2) 52°; 3) 172°?
16. 3HaHAiTb KYT, HKID;O H:oro 6iceKTpHca YTBOpI06 ia CTOPOHOIO
KyT, .llKHH ,lOpiBHI06! 1) 60°; 2) 75°; 3) 89°.
17. ,lJ;oBeAiTb, w;o 6iceKTpHca KyTa YTBOpl06 3 H:oro CTOPOHaMH KYTH
He 6iJILmi aa 90°.
18~ ,lJ;oBeAiTb, w;o KOJIH npoMiHb BHXOAHTb 3 BepmHHH KyTa i YTBO-
PI06 3 H:oro CTOPOHaMH piBHi rocTpi KYTH, TO BiH 6 6iceKTPHCOIO
KyTa.
19. 3HaiiAiTL KYT Mi>K 6iceKTPHCaMH cyMi)KHHX KyTiB.
20. ,lJ;oBeAiTb, w;o 6iceKTpHCH BepTHKa.JibHHX KyTiB JiemaTb Ha
OAHiH np.RMiH.
0 A
Man. 42 Man. 43

7 KJiac 32
21. 3Haii,zi;iTL KYT Mim 6iceKTpHco10 i npo.zi;oB>Kemu1M o.zi;Hie1 ia
CTOpiH .zi;aHOl'O KyTa, SKHH .zi;opiBHIOG: 1) 50°; 2) 90°; 3) 150°.
22~ 3 BepIIIHHH 0 cyMi>KHHX KYTiB AOB i COB npoBe.zi;eHo npoMiHL
OD y niBnJiow;HHy, .zi;e npoxo.zi;uTL cniJILHa cropoHa KYTiB OB
(MaJI. 43). ,ll;oBe.zi;iTL, w;o npoMiHL OD nepeTHHae a6o B~piaoK
AB, a6o Bi.n;piaoK BC. HKHii a Bi.zi;piaKiB nepeTHHae npoMiHL OD,
a~o KYT AOD MeHmHii (6iJILmHii) Bi.n; KyTa AOB? IloacHiTL
Bi,ll;IlOBi.zi;L.
23. 3 BepmHHH poaropHyToro KyTa (aa1) B o.zi;Hy niBnJiow;HHy
npoBep;eHo npmd b i e. "lJ.oMy .n;opiBHIOe KYT (be), ~rnw;o:
1) L (ab)= 50°, L (ae) = 70°; 2) L (a1b) = 50°, L (ae) =
= 70°; 3) L (ab)= 60°, L (a1e) = 30°?
24. 3 BepmHHH poaropHyToro KyTa (aa1), B O,ll;HY niBilJIOlll;HHY npo-
Be.zi;eHO npoMeHi bi e. Bi.zi;oMO, w;o L (ab)= 60°. L (ac) c= 30°.
3Haiip;iTL KYTH (a1b), (a1e), i (be).
25. Bi.zi; niBnpaMo1 AB y piaHHx niBnJiow;HHax BiAKJia,n;eHo KYTH
BAC i BAD. 3Haii.zi;iTL KYT CAD, rucw;o: 1) L BAC = 80°,
L BAD= 170°; 2) L BAC= 87°, L BAD= 98°; 3) L BAC=
= 140°, L BAD= 30°; 4) L BAC = 60°, L BAD= 70°.
26~ ,ll;aHo TPH npoMeHi a, b i e ia cniJILHHM 11011aTKOM. Bi.n;oMo, w;o
L (ab)= L (ae) = L (be)= 120°. 1) "lJ.u npoxo,n;HTL o.zi;HH
3 ~HX npoMeHiB Mi>K CTOpoHaMH KyTa, YTBOpeHoro ,ll;BOMa
iHmHMH npoMeHSMH? 2) liH MO>Ke qpaMa nepeTHH8TH yci TpH
npoMeHi? IIoacHiTL Bi.n;noBi.zi;L.
§ 3. 03HAKH PIBHOCTI TPHKYTHHKIB
20. IIEPIDA 03HAKA PIBHOCTI TPHKYTHHKIB
Teo p e Ma 3.1 (oaHaKa piBHOCTi TpHKYTHHKiB aa ,n;BoMa
CTOpOH8MH i KYTOM Mi>K HHMH). H1Cll'10 iJBi CTOpoHU i xyr JI/CW HU.MU
oiJnozo rpuxyrnuxa iJopiBH'KJJOTb siiJnosiiJno iJBoM cropoHaM i xyry
MUe HU.MU iJpyzozo TpUICYTHUJCa, TO raJCi rpuxyTHUICU piBHi.
,21; o B e .zi; e H H a. Hexaii y TPHKYTHHKiB ABC i A 1B1C1 AB =
= A1Bi, AC= A1C1, LA= L Ai. ·,ll;oBe.zi;eMo, w;o TPHKYTHHKH
piBHi (MaJI. 44).
Hexaii A 1B2C2 - TpHKYTHHK, w;o .zi;opiBHIOG TPHKYTHHKY ABC,
i iioro BepmHHa B2 ne>KHTb Ha npoMeHi A 1Bi, a BepmHHa C2 - y Tiii
caMiii niBnJIOlll;HHi Bi,ll;HOCHO npaMo1 A 1Bi, .zi;e Jie:>KHTL BepmHHa C1

t 3, 03HllKH piBHOCTi TpHKYTHHKiII
A 8
Man. 44
A'1
33
(MaJI. 45, a). OcKinLKH A 1B1= A 1B2, TO :eepmHHa B2 a6iraeTLCa
a :eepmHHoIO B1 (Man. 45, 6). qepea Te m;o L B1A1C1 = L B2A1C2,
npoMiHL A1C2 a6iraeTLCa a npo11eue11 A1C1 (MaJI. 45, 6). OcKiJILKH
A1C1 = A1C2, TO sepmHHa C2 a6iraen.ca a sepmHHOIO C1 (MaJI.
45, z). Orme, TpHKYTHHK A1B1C1 a6ira8'l'ltca a TPHKYTHHKOM A1B:£2,
T06TO ,11;opiBHI08 TPHKYTHHKY ABC. TeopeMy ,11;ose,11;eHo.
If 3 a p, a 11 a (1). Bi,n;piaKH AB i CD nepeTHHaIOTLCa B TO'l-
V ~ 0, RKa e cepep,RBOIO KOJKHOro a HHX. qoMy p,opiBBJoe si,ll;pi-
aoK BD, aK:m;o si,ll;piaoK AC= 10 M?
Po a B' a a a Ha a. TpHKYTHHKR AOC i BOD piBHi aa nep-
mo10 03HaKOIO piBHOCTi TpBKYTHJll(iB (JIAJI. 46). y HHX KYTH
A, 81 82
A, 8,(82)
a)
~[2
o)
/I C,{C2)
I
I
I
I
I
I
A, 81 (82) A, B1(Bz)
8) 2)
Man. 45
2 feoMeTpiR, 7-9 Kn.

7 KJI8C
[
MaJI. 46
D
34
AOC i BOD piuHi. SIK uepTHKaJILHi,
a OA = OB i OC = OD, TOMY ~o To11Ka
0 - cepeAHHa uiApiaKiB AB i CD. 3 piu-
HocTi TPHKYTHHKiB AOC i BOD BH-
rnIHuae piuHiCTL cTopiH AC i BD.
OcKiJILKH aa yMOBOIO aaAa'li AC= 10 M,
TO i BD = lOM.
21. BHKOPHCTAHHH AKCIOM IIPH AOB:EAEHHI TEOPEM
HK MH aHaGMO, niA qac AOBe,ll;eHHSI TeopeM A03BOJIHGTLCSI Kopuc-
TyBaTHCSI aKcioMaMH i paHime AOBeAeHHMH TeOpeMaMH. HK npauH-
JIO, y npo~eci AOBeAeHHSI IlOCHJiaIOTLCH He Ha HOMep aKCiOMH aa
cnucKoM, a Ha 'ii aMiCT. CaMe TaK 6yJio apo6JieHo, KOJIH AOBOAHJIH
nepmy oaHaKy piBHOCTi TPHKYTHHKiB (TeOpeMa 3.1). PoarJIHHeMo
~e paa ~e AOBeAeHHH, Ha3HBaIO'IH aKciOMH, HKi 6yJIO BHKOPHCTaHO.
,ZJ;oBeAeHHSI no11HHaeTLCH cJIOBaMH: •Hexaii A 1B£2 - TPHKYT-
HHK, ~o ,11;opiBHIOG TPHKYTHHKY ABC, a uepmHHOIO B2 Ha npoMeHi
A1B1 i uepmHHoIO C2 y Tiii caMiii niBnJio~HHi BiAHocHo npSIMO'i
A1B1, Ae JiemHTL uepmHHa C1•. TaKHii TPHKYTHHK, S1K MH aHaeMo,
icHye aa aKcioMoIO VIII.
,ZJ;aJii CTBepAmyeTLCSI, ~o uepmHHH B1 i B2 a6iraIOTLCSI Ha Tiii
niAcTaui, ~o A 1B1 = A.1B2. TyT BHKopHcTaHo aKcioMy BiAKJiaAaHHH
BiApiaKiB (aKcioMa VI).
IIOTiM CTBePA*YGTLCSI, ~o npoMeHi A 1C2iA1C1 Tem a6ira10TLCS1,
TOMy ~o L B1A1C1 = L B2A1C2. TyT BHKOpHcTaHo aKcioMy BiA-
KJiaAaHHSI KyTiB (aKcioMa VII).
HapemTi CTBepAmyeMo, ~o ii BepmHHH C1 i C2 a6iraIOTLCH,
ocKi.JtLKH A 1C1 = A£2. TyT 3HOBY BHKopucTaHo aKcioMy VI.
Ba'IHMO, ~o AOBeAeHHSI TeopeMH 3.1 rpyHTYGTLCH JIHme Ha
aKcioMax.
22. ~PJ'TA 03HAKA PIBHOCTI TPHKYTHHKIB
Teo p e Ma 3.2 (oaHaKa piBHOCTi TPHKYTHHKiB aa cropoHoIO
i npHJierJIHMH ,11;0 He'i KYTaMH). HKUfo C'l'Ol'OH4 i npu11.ezir.i iJo Hei
xyTu oiJHOZO TpuxyTHUlUI, iJopi6H'KJ10T"b 6iiJno6i8HO CTopoHi unpu-

§ 3. OaHaKH piBHoCTi TPHKYTHHKiB 35
A 8
MaJI. 47
MZJIUM iJo He'i 1'YTa.M iJpyzozo TpU1'YTHUKa, TO T41'i rpU1'YTHU1'U
pitmi.
AoBe/1,eHHS. Hexaii ABC i AiBiCi -/1,Ba TPHKYTHHRH,
y RKHX AB= AiBi. .LA= L Ai i LB= L Bi (Man. 47).
Aose/1,eMo, mo TPHKYTHHKH piBHi.
Hexaii A iB£2 - TPllKYTHHK, mo /1.0piBHIOe TPHKYTHHKY ABC, i
ii:oro sepmHHa B2 nemHT:& ua npoMeHi AiBi. a sepmuua C2 -y Tm
caMiii: niBnnomHHi si/1,uocuo npsMoi A iBi. n.e nemHT:& sepmHHa
Ci.
OcKiJI:&KH A iB2 = A iB1, TO sepmHHa B2 a6iraeT:&Cs a sepmuuoio
Bi. qepea Te mo L BiAiC2 = L BiAiC1 i L AiB1C2 = L AiB1Ci.
npoMiH:& AiC2 a6iraeT:&es a npoMeHeM AiCi, a npoMiH:& BiC2 - a
npoMeHeM B1C1. 3Bi/1,CH BHUJIHB86, mo Bepmuaa C2 a6iraeT:&eS 3
BepWHHOIO CI·
0Tme, TPHKYTHHK AiB1Ci a6iraeT:&cs a TPHKYTHHKOM A iB£2,
To6To /1,opisa10e TPHKYTHHKY ABC. TeopeMy /1,0Be/1,eao.
23. PIBHOB:EWEHHB TPHKYTHllK
TpHKYTHHK H83HB86T:&CS pitmo6eopenUM. RI<mo B H:&Ol'O /I.Bi CTO-
poHH piBHi. Qi piBHi CTOPOHH Ha3HB8IOT:&CS 6i'l.HUMU croponaMU, a
Tj)e'l'S CTOpoHa Ha3HB86T:&CS OCH06010 TPHKYTHHKa.
Ha MaJIIOHKY 48 ao6pameao piBHOOe/1,peHHH TPHKYTHHK ABC.
Y H:&0ro 6i11ui cTopoHH AC i BC, a ocuoBa AB.
Te ope M a 3.3 (BJI&CTHBicn KYTiB piBuo6e~peaoro TpHKYTHH-
Ka). Y JR,sH06eiJpe1uu&y rpurcyrHU.icy icyru npu ocHOtli pitmi.
Ao Be /1. e H H s. Hexaii ABC - piBHo6e/1,peHHii TPRKYTHHK
a ocaosoio AB (MaJI. 48). AoBe/1,eMo, mo B H:&Oro LA= LB.

7 KJiac 36
[ [
A A
MaJI. 48 MaJI. 49
TpHKYTHHK CAB AOPiBHIOe TPHKYTHHKY CBA aa nepwo10 oaaa-
KOIO piBHOCTi TPHKYTHHKiB. CnpaBAi, CA = CB, CB = CA, L C =
= LC. 3 piBHocTi TpHKYTHHKiB BHITJIHBae, ~o LA= LB.
TeopeMy AOBeAeHo.
TpHKYTHHK, y 11Koro BCi cTopoHH piBHi, aaaHBa6TbCH piBHOCTo-
poHHiM.
3 a A a 'l ~ (12). ~oB~AiTh, ~o y piBHOCTOpoHHbOMY TpH-
KYTHHKY BCl KYTH PlBHl.
Po a B' 11 a a H H H. Hexaii ABC - AaHHii TPHKYTHHK
a piBHHMH cTopoHaMH: AB= BC= CA (MaJI. 49). OcKiJILKH
AB = BC, TO ~eii TpHKyTHHK piBHo6eApeaHii a OCHOBOIO
AC. 3a TOOpeMoIO 3.3 L C = L A. 'lepea Te ~o BC = CA,
TPHKYTHHK ABC piBH06eApeHHii a OCHOBOIO AB. 3a TeopeMoIO
3.3 L A = L B. TaKHM 'lHHOM, L C = L A = L B, To6To
BCi KYTH TPHKYTHHKa piBHi.
24. OBEPllEHA TEOPEMA
Teo p e Ma 3.4 (oaaaKa piBao6eApeHoro TPHKYTHHKa). H-,c~o
6 'l'pUKY'l'HUKY iJBa 'ICY'l'U pi6Hi, 'l'O BiH pi6HOOeiJpeHUU.
~ o Be A e H Ha. Hexaii ABC - TPHKYTHHK, y HKoro L A =
= L B (MaJI. 50). ~oBeAeMo, ~o BiH pisao6eApeHHii a ocaoso10 AB.
TpuKyTHHK ABC AOPiBHI06 TPHKYTHHKY BAC aa APYro10 oaHa-
KOIO piBHOCTi TPHKYTHHKiB. CnpaBAi, AB= BA, LB= LA,
L A = L B. 3 pisaocTi TPHKYTHHKiB BHITJIHBas, ~o AC = BC.
OTme, aa oaaa'leHHHM TPHKYTHHK ABC - pisao6eApeHHii. TeopeMy
AOBeAeHo.
TeopeMa 3.4 Ha3HBa6TbCH o6epHen010 AO TeopeMH 3.3. BucHoBoK
TeopeMH 3.3 e YMOBOIO TeopeMH 3.4, a YMOBa TeopeMH 3.3 6 BHCHOB-

§ 3. OaHaKH piBHOCTi TJlHKYTHHKiB
KOM TeopeMH 3.4. He KO)l(Ha TeopeMa Mae
o6epHeHy, ro6To .HK~o AaHa TeopeMa npa-
BHJILHa, TO o6epHeHa TeopeMa Mome 6YTH He-
npaBHJILHOIO. Ilo.HCHHMO u;e Ha npHKJI&Ai
TOOpeMH npo eepTHKaJILHi KYTH. D;io reopeMy
MO)l(Ha CcPOPMYJIIOB&TH TaK: .HK~O AB& KYTH
BeJ)THKaJILHi, TO BOHH piBHi. 06epHeHa AO Hei'
TOOpeMa Mana 6 6YTH TaKOIO: .HK~O ABa KYTH
37
piBHi, To BOHH BepTHK&JILHi. A u;e, 3BH11aiiHo, A B
HenpaBHJILHO. ,lJ;Ba piBHi KYTH 30BCiM He Mair. 50
o6oB'.H3KOBO IlOBHHHi 6YTH eepTHKaJILHHMH.
t1 3 a A a 11 a (16). Ccl>opMynioiiTe i AOBeAiTL TeopeMy, o6ep-
V HeHy AO TBepAmeHH.H 3aAatti 12.
Po 3 B' H 3 a H H H. YMoea 3aAa'li 12 noJIHrae e TOMy, ~o
TpHKyTHHK piBHOCTOpoHHiH, a BHCHOBOK - BCi KYTH TpHKYT-
HHKa piBHi. ToMy o6epHeHa TeopeMa noBHHHa cl>opMynioeaTH-
c.H TaK: .HK~O y TPHKYTHHKa eci KYTH piBHi, TO BiH piBHOCTO-
pOHHiH.
,D;oeeAeMo u;io TeopeMy. Hexaii ABC - TPHKYTHHK 3 piBHH-
MH KyTaMH: LA = L B = L C. OcKiJILKH L A = L B, To
3a TeopeMoIO 3.4 AC = CB. OcKiJILKH L B = L C, TO aa
TeopeMoIO 3.4 AC = AB. TaKHM 'IHHOM, AB = AC = CB,
TOOTO yci CTOpoHH TpHKYTHHKa piBHi.
OTme, 3a oaaa11eHHS1M TPHKYTHHK ABC - piBHocTOpoHHiii.
25. BHCOTA, BICEKTPHCA I M:EMAHA TPHKYTHHKA
Bucoro10 TPHKYTHHKa, on~eHOIO a AaHoi' eeplIIHHH, HaaHBae-
TLC.H nepneHAHKYmq1, npoBeAeHHii 3 u;iei' BepmHHH AO np.HMOi', ~o
MiCTHTL npoTHJiemHy CTOpoHy TPHKYTHHKa.
Ha MaJIIOHKY 51BH6a11HTe ABa TPHKYTHHKH, y .HKHX npoeeAeHo
BHCOTH a BepmHH Bi B1. Ha MaJIIOHKY 51, a ocHoBa BHCOTH nemHTL
Ha CTOpoHi TPHKYTHHKa, Ha MaJIIOHKY 51, 6 - Ha llPOAOB)l(eHHi
CTOpOHH TpHKYTHHKa.
Bice1Crpuco10 TpHKYTHHKa, npoeeAeHoIO 11 AaHoi' eepmHHH,
HaaHBaeTLC.H eiApiaoK 6iceKTpHCH KyTa TPHKYTilHKa, ~o cnony11ae
D;IO eepwHHy 3 TO'IKOIO Ha npoTHJiemHiH CTOpOHi (MaJI. 52, a).

7 KJiac 38
B,
A [ A, c,
a) tJ)
MaJI. 51
[ A
tJ)
[
a)
Man. 52
Meoiano10 TPHKYTHHKa, npoae,D;eHOIO a ,D;aHo'i aepmHHH, H&3HBae-
TLCSI Bi,D;piaoK, ~o cnonyqae ~ sepmHHy i3 cepeAffHOIO npoTHnem-
uo'i CTOpOHH TPHKYTHHK& (Man. 52, 6).
26. BJIACTHBICTL ~AHR PIBBOBEWEHOI'O
TPHKYTHBKA
Te ope M a 3.5 (BJI&CTHBicTL Me,D;iaHH piBH06e,D;peHoro TpH-
KYTHHKa). y piBHo6ei1-peHOMY rpUIC'flTHUIC'fl .-eiJiaHa, npOBeiJeHa iJo
OCHOtlU, e 6icnrpuco10 i tsucoro10•
.u; o a e A e H H 11. Hexaii ABC - ,D;&HHH .piBHo6e,D;peHHii TpH-
KYTHHK 3 ocuoao10 AB, a CD - Me,D;iaHa, npoae,D;eHa ,D;O OCHOBH
(MaJI. 53).
TpHKYTffHKH CAD i CBD piaui aa nepmo10 oaHaKoIO piBHOCTi
TPHKYTHHKiB. (Y RHX CTOpoRH AC i BC pi.Bui, 6o TPHKYTHHK ABC
piB&o6e,D;pead. KYTH CAD i CBD pi.Bui SK KYTH npH ocaoBi piBH0-
6e,D;peHoro TPHKYTHHKa. CTopoHH AD i BD piBHi, OCKiJibKH D -
cepe,ZJ;HHa Bi,ZJ;piaKa AB.)

§ 3. OaHaKH piBHOCTi TJ>HKYTHHKiB 39
l [
A 8 A B
MILll. 53 MaJI. 54
3 piBHOCTi TpHKYTHHKiB BHIIJIHBae piBHicTb KYTiB: L ACD =
- L BCD, L ADC = L BDC. OcKiJILKH KYTH ACD i BCD piBHi,
TO CD - 6iceKTPHCa. "llepea Te ~o KYTH ADC i BDC CYMimHi
i piBHi, TO BOHH np&Mi, TOMY CD - BHCOTa TPHKYTHHKa. TeopeMy
,ll;OBep;eHO.
3 a A a 11 a (28). ,IJ;oBep;iTL, ~o 6iceKTpHca piBHo6~,zi;peHor~
TPHKYTHHKa, npoBep;eHa 3 BepmHHH, npoTHJiemHOI OCHOBl,
6 Me,zi;iaHOlO i BHCOTOJO.
Po a B' sr. a a H H s. Hexaii ARC - piBHo6ep;peHHii TpH-
KYTHHK a OCHOBOJO AB i CD - H.oro 6iceKTpHca (MaJI. 54).
TpHKyTHHKH ACD i BCD piBHi aa nepmo10 oaaaKoJO: y HHX
cropoHa CD cniJILHa, cropoHH AC i BC piBHi SJ.K 6i'IHi cropoau
piBHo6ep;peHoro TPHKYTHHKa, a KYTH npH BepmHHi C piBHi
TOMy, ~o CD - 6iceKTPHCa. 3 piBHocTi TPHKYTHHKiB BHnJIH-
Bae piBHiCTL ix cTopia AD i BD. 0Tme, CD - Mep;iaaa TPHKYT-
HHKa ABC. A aa BJJaCTHBiCTJO Me,zi;iaHH piBHo6ep;peaoro
TPHKYTHHKa BOHa 6 i BHCOTOIO.
27. TPET.H 03HAKA PIBHOCTI TPHKYTHHKIB
Teo p e Ma 3.6 (oaHaKa piBHocTi TPHKYTHHKiB aa TPLOMa cTo-
poHaMH). H'l«lfo 7pu C'l'OpoHU oiJHoio 7puxyrHuxa iJopisHJOIOl'b BiiJ-
noBWHO 7pbOM C'l'OpoHaM iJpyioio 7puxyrHwca, 'l'O ra.JCi rpu-,cyr-
HUKU piBHi.
,n; o Be Ae H H sr.. Hexaii ABC i A 1B1C1 - p;Ba TPHKYTHHKH,
y SJ.KHX AB= A1Bi, AC= A1Ci, BC= B1C1 (MaJI. 55). Tpe6a
AOBecTH, ~O TPHKYTHHKH piBHi.
IIpHnyCTHMO, ~o TPHKYTHHKH He piBHi. Top;i y HHX L A =I=
=I= L Ai, LB =I= L B1, LC =I= L C1. laaKme BOHH 6yJIH 6 piBHi aa
nepmo10 oaHaKoIO.

7 KJiaC 40
[
A B B,
MaJI. 55
Hexaii A1B1C2 - TpHKYTHHK, w;o .a;opiBHI06 TPHKYTHHKY ABC,
a BepmHHa iioro C2 JiemHTb B o,o;HiH niBnJIOIIJ;HHi a BepmHHOIO C1
B~HOCHO npRMo1 A1B1 (MaJI. 55).
Hexa:H D - cepe,o;HHa s~pi3Ka C1C2. TpHKYTHHKH A1C1C2 i
B1C1C2 ·- piBHo6eApeHi i3 cniJibHOIO ocHosoro C1C2. ToMy 1x
MeAiaHH A 1D i B 1D e BHCOTaMH. To.a;i npRMi A 1D i B 1D nepneH,ll;HKY-
m1pHi ,o;o npRMo'i C1C2. IIpHMi A1D i B1D He a6iraIOTbCR, ocKiJibKH
TO'IKH Ai. B1, D He JiemaTb Ha OAHiH npSIMiH. AJie 11epea TO'IKY D
npSIMo'i C1C2 MomHa nposecTH JIHme o,o;Hy nepneHAHKYJISIPHY .a;o He'i
npSIMY. MH AIBIIlJIH ,o;o cynepe11HocTi. TeopeMy .a;ose,o;eHo.
3a,11;a11a (29). Y _TPHKYTHHKiB ABC i A1B1C1: AB=
= A1B1, AC= A1Ci. LC= L C1 = 90°. ,ll;oee,o;iTb, w;o
6. ABC= 6. A1B1C1.
Po 3 s' Ra a H HSI. Hexa:H ABC i A 1B1C1- AaHi TPHKYT-
H:KKH (MaJI. 56). 1Io6YAY6MO TpHKYTHHK CBD, w;o ,o;opiBHIOG
TPHKYTHHKY CBA, i TpHKYTHHK C1D1B1, w;o ,o;opiBHIOe TpH-
KYTHHKY C1A1B1.
8 B,
A [ D A, c, o,
MaJI. 56

§ 3. 03HRKH pillHOCTi TPHKYTHHKill 41
TpeKyTHHKH ABD i A1B1D1 piBHi aa TpeThOIO oaHaKoIO.
Y HHX AB= A1Bi. 3a yMOBOIO 3&Aa•li; AD= A1Di. ocKiJihKH
AC= A1C1; BD = B1Di. 'lepe3 Te ~o BD =AB, B1D1 =
= A1B1. 3 piBHoCTi TpHKYTHBKiB ABD i A1B1D1 BHnJIHBas
piBHicTh KyTiB: L A = L A 1. OcKiJxbKH 3a yMOBOIO AB =
= A1Bi. AC= A1Ci. a LA= L A1 38 AOBeAeHHM, To
TPHKYTHHKH ABC i A1B1C1 piBHi 3a nepmo10 03H8KOIO.
28. .HK CAMOCTIHHO rOTYBATHCb 3A Ill,ll,PY'IHHKOM
Mome TpanHTHCJ., ~o 3 ,u;eaKHX npH'IHH, H8IlPHKJiaA 11epea
XBOpo6y, BH He 6yJIH Ha ypo~ reoMeTpii. To,u;i M8TepiaJI :u;i.oro ypo-
KY B&M AOBeAe'l'bCS BHB'IHTH caMocTiiiHo aa niApy'IHHKOM. TeKcT
ni,D;py'IHHK8 CJii.A 'IHT8TJI He nocrrimaIO'IH, OKpeMHMH pe'leHHBMH,
He uepeXOAHTH AO H8CTYilHOI'O pe11eHHa, He 3po3YMiBWH 3MiCT
nouepeAHi.oro. PoarJIHHeMo KOHKpeTHHii rrpHKJia,u; - ,u.oBe,u.eHHSI
TpeThOi oaaaKH piBHOCTi TPHKYTHHKiB. 0Tme, 'IHTaeMo TeKCT ni,u.-
PY'IHHKa:
•.HK~o TPH CTOPOHH o,u.aoro TpHKYTHHKa AOPiBHIOIOTb Bi,u.noBiA-
HO TphOM CTopoHaM ,u.pyroro TPHKYTHHKa••••
~o6 3po3yMiTH 3MiCT ~Oro pe'IeHHR, Tpe6a 3HaTH, w;o T&Ke TpH-
KYTHHK, Aoro cTopoHH i piBHiCTi. cropiH. BH Bee ~e aaaere, TOMY
3Mi.CT rrpo'IHTaHOl'O pe11eHH$1 BRM apoayMiJIHii. qHT86MO ,U.8Jli:
tTO TRKi TpHKYTHHKH piBHh.
~o6 3p03YMiTH 3Mi.CT ~ro pe'IeHHH, rrOTpi6HO 3H8TH, aKi
TpmcyTHHKH H&3HB&IOTbCH piBHHMH. A.ne BH i :u;e 3Ha6Te. TaKHM
'IHHOM, 3MiCT TeopeMH BaM 3po3yMiJIHH. qHTa6MO AOBeAeHHa.
;lJ;oBe,u.eaaa. •Hexaii ABC i A1B1C1 -,u.Ba TPHKYTHHKH,
y aKHX AB= A1Bi. AC= A1Ci. BC= B,c, (MaJI. 55). IIoTpi6ao
,11,0BecTH, w;o TpHKYTHHKH piBHh.
TYT Bee apoayMiJio. IloaHa'IaIOTl>ca TPHKYTHHKH, HKi 38,11,0BOJih-
HaIOTh yMoBy reopeMH i piBHiCTb HKHX noTpi6Ho ,u.oBeCTH.
•IIpHilYCTHMO, w;o TpHKYTHHKH He piBHi•.
Bu 68'IHTe, w;o po6HMO npunyw;eHHH, rrpoTBJiemHe TBepAmeHHJO
TeOpeMH. OTme, B npo:u;eci no,u.an:i.DIHx MipKyB&Hb MH M86MO npH-
iiTH 11.0 cynepe'IHOCTi (AOBe,u.eHHa Bi.A cynpoTHBHoro).
•To,u.i y HHX LA =I= L Ai, LB =I= LB,, LC =I= LC1.
IHaKme BOHH 6yJIH 6 piBHi 38 nepmo10 oaaaKOIO•.

7 KJiac 42
IlpHra,11;aiiTe nepmy oaHaKy piBHOCTi TPHKYTHHKiB. IIepeKoHaii:-
TecL y TOMY, m;o KOJIH BHKOHy6TLCS xoqa 6 O)J;Ha 3 piBHOCTeH
LA= L Ai, LB= L Bi, LC= L Ci, TO TPHKYTHHKH ABC i
A1B1C1 pisHi, a u;e cynepeqffT& apo6JieHOMY npHnym;eHHIO.
•Hexaii A1B1C2 - TPHKYTHHK, m;o ,11;opiBHI06 TPHKYTHHKY ABC,
i iioro BepmHHa c2 Jie)l(HTL B O,ll;HiH niBilJIOID;HHi 3 BepmHHOIO cI
Bi,ll;HocHo npaMo'i A 1B1 (MaJI. 55)•.
TYT see apooyMiJio. We10 cl>paao10 noqHHaJIOCL ,11;ose,11;eHHH nep-
mo'i i ,11;pyro'i oaHaK.
•Hexaii D - cepe,IJ;HHa si,ll;pi3Ka C1C2•.
BH 3Ha6Te, m;o TaKe cepe,11;HHa si,ll;pi3Ka.
•TpHKYTHHKH A1C1C2 i B1C1C2 - pisHo6e,11;peHi ia cniJILHOIO
OCHOBOJO C1C2•.
~o6 3po3yMiTH 3MiCT D;LOl'O TBep,11;meHHS, Il0Tpi6HO 3HaTH, JIKHH
TPHKYTHHK Ha3HBa6TLCS piBHo6eApeHHM i JIKa iioro CTOpoHa Ha3H-
Ba6T&CS OCHOBOIO.
•ToMy 'ix Me,11;iaHH A 1D i B1D e BHCOTaMH•.
3MicT o;LOro peqeaaa BaM apooyMiJIHii. BH aaaeTe, m;o TaKe
Me,ll;iaHa, BHCOTa, i 3HaeTe BJiaCTHBiCTb Me)J;iaHH piBH06e,ll;peHOl'O
TPHKYTHHKa.
•OTme, npBMi A 1D i B1D nepneH,ll;HKyJiapHi ,11;0 npsMol C1C2•.
HcHo.
•IlpHMi A1D i B1D He a6iraJOTLCH, ocKiJILKH TO'IKH Ai. Bi. D He
JiemaTL Ha O,ll;Hiii npHMiiit.
3poayMiJio. HK6H TO'IKa D JiemaJia Ha npaMiii A 1Bi. TO TO'IKH
C1 i C2 JiemaJIH 6 y piaHHx niBIIJIOID;HHax BiAHOCHo npsMo'i A 1B1.
•AJie 11epea T011Ky D npBMo'i C1C2 MOmHa nposecTH JIHme OAHY
nepneH,ll;HKYJISPHY ,11;0 He'i npHMY•·
Hcao. BH aHaeTe TaKy TeOpeMy.
•MH npHHmJIH ,11;0 cynepe'IHOCTb.
3pooyMiJIO.
• TeopeMy ,11;0BeAeHo•.
? KOHTPO~bHI3AilHTAHHH
•1. )J;oBeAiTL nepmy oaHaKy pisHocTi TpHKYTHHKiB. HKi aKcioMH
BHKOpHCTOBYIOTLCS npH ,ll;OBe,ll;eHHi TOOpeMH 3.1?
2. Ccl>opMyJ110iiTe i ,ll;OBe,ll;iTb Apyry oaHaKy piBHocTi TPHKYTHHKiB.
3. SIKHH TPHKYTHHK Ha3HBaGTLCJI piBHo6e,11;peHHM? SIKi CTOPOHH

§ 3. 0aHaJCH piBHOCTi TPHKYTHBKiB 43
piBHOOeApeHoro TPHKYTHHKa HR3HBRIOTbCSI 6i'IHHMH CTOpo-
HRMH? Hxa CTOpOHa HR3HBR6ThCSI OCHOBOIO?
4. ,IJ;oBeAiTb, m;o y piBH06eApeHOMY TPHKYTHHKY KYTH npH OCHOBi
piBHi.
5. HKHii TpHKYTHHK Ha3HBR6TbCSI piBHOCTopoHHiM?
6. ,IJ;oBeAiTb, m;o KOJIH y TpHKYTHHKY ,D;Ba KYTH piBHi, TO BiH
piBHOOe,ZpeHHH.
7. IloSicHiTb, m;o TaRe o6epHeHa TeopeMa. Haue,ll;iTb npHKJiaA. 1IH
)'.JI$l KOlRHOl TeopeMH npaBHJibH8 o6epHeHa?
8. ~o T&Ke BHCOTa TPHKYTHHK8?
9. ~o T&Ke 6icexTpHC8 TPHKYTHHK8?
10. ~o TaKe Me,ll;iaaa TpHKYTHHK8?
11. ,D;ose,ll;iTb, m;o y pisuo6eApeHoMy TPHKYTHHKY Me,ZJ;iaHa, npose-
,ZJ;eaa ,Zl;O OCHOBH, e BHCOTOIO i 6iceKTpHCOIO.
12. ,IJ;oae,ll;iTb TPeTIO oaHaKy piBHocTi TPHKYTHHKiB•
• 3A~'II
1. BiApiaxH AB i CD nepeTHHaIOTbC.H B ro11~ 0, SIKa e cepeAHHOIO
KO>KHoro a uux. 1IoMy ,D;opiBHI06 ui,lpiaox BD, HKID;O AC=
= 10M?
2. 1Iepea cepe,ZJ;HHy 0 Bi,lpiaKa AB npoBe,D;eHo npSIMY, nepneHAHKY-
JI.HPHY AO np.HMoi AB (MaJI. 57). ,IJ;oue,ll;iTb, m;o KOlRHa TO'IKa ~iei
np.HMoi O.ZH&KOBO Bi.lA&JieHa um TO'IOK A i B.
3. Ha cTopoHi AB TPHKYTHHKa ABC B3HTo TO'IKY D, a ua cTopoui
A1B1 TPHKYTHHKaA1B1C1 - T011Ky D1. Bi,loMo, m;o TPHKYTHHKH
ADC i A1D1C1 piBHi ii Bi,D;piaKH DBi D1B1 piBHi. ,IJ;oue,ll;iTb piB-
HicTb TPHKYTHHKiB ABC i A1B1C1:
A 0 B
£ D
MaJI. 57 MaJI. 58

7 Knac 44
4. ID;o6 BHMip$1TH Ha Micu;eBOCTi Bi,ll;CTaHb Mim ,ll;BOMa TO'IKaMH
A i B, Mim 8KHMH He MOmHa npoiiTH no np8Miii (MaJI. 58), BH6H-
paIOTb TaKy TO'IKY c, 3 $1KOl MO)KHa npoHTH ,11;0 TO'IKH A i ,11;0
TO'IKH Bi a 8Koi BH,n;Ho o6H,n;Bi u;i TO'IKH. IlpoBiwyIOTL 1Bi,n;cTa-
Hi AC i BC, npo,n;oBmyIOTb ix aa TO'IKY C i Bi,ll;KJia,n;a10T1> CD =
= AC i EC= CB. To,n;i Bi,n;piaoK ED ,n;opiBHIOe wyKaHiH Bi,n;-
cTaHi. Il0$1CHiTb 'IOMy.
5. Bi,ll;piaKH AB i CD nepeTHHaIOTbC$1 B TO'lu;i 0 (MaJI. 59). ,ll;oBe-
,n;iTL piBHiCTb TPHKYTHHKiB ACO i DBO, KOJIH Bi,n;oMo, w;o KYT
ACO ,n;opiBHI06 KYTY DBO i BO= CO.
6. Bi,n;piaKH AC i BD nepeTHHaIOTLC$1 B TO'ID;i 0 (MaJI. 60). ,ll;oBe-
,n;iTh piBHiCTb TpHKYTHHKiB BAO i DCO, KOJIH Bi,n;oMo, w;o KYT
BAO ,11;opiBHI06 KYTY DCO i AO = CO.
7"!' ,ll;oBe,ll;iTb piBHiCTb TPHKYTHHKiB aa Me,11;iaHOIO i KyTaMH, Ha $1Ki
BOHa poa6HBa6 KYT TpHKYTHHKa.
8. ID;o6 BHMip8TH Ha Micu;eBOCTi Bi,ll;CTaHb Mim ,ll;BOMa TO'IKaMH A i
B, a $1KHX o,11;Ha (To'IKa A) HenpHcTynHa, npoBiwyIOTb Hanp8M
Bi,ll;piaKa AB (MaJI. 61) Ta Ha :Horo npo,11;0BmeHHi Bi,n;KJia,n;aIOTb
,n;oBiJihHHH Bi,ll;piaoK BE. B116Hpa10T1> Ha Micu;eBocTi TO'IKY D,
a 8Koi BH,ll;HO TO'IKY A i MO»eHa npoiiTH ,n;o TO'IOK Bi E. Ilpo-
. Biwy10T1> np8Mi BDQ i EDF i BHMip1010T1> FD = DE i DQ =
= BD. ,ll;ani ii,11;yT1> ya,n;oBm np8Moi FQ, ,ll;HB1l$1'1HCb Ha TO'IKY
A, noKH He aHaii,n;yT1> TaKy TO'IKY H, 8Ka JiemHTb Ha np8MiH AD.
To,n;i HQ ,n;opiBHIOe wyKaHiH Bi,ll;cTaHi. ,ll;oBe,n;iTL u;e.
9. IlepHMeTp (cyMa ,ll;OB»CHH CTOpiH) piBH06e,n;peHoro TpHKYTHHKa
,n;opiBHIOE: 1 M, a OCHOBa 0,4 M. 3Haii,n;iTb ,ll;OB)KHHY 6i'IHOl
CTOpOHH.
A D A 0
Man. 59 Man. 60
1
HanpRM BHaHa'laJOT& *ep,ll;HH&MH-eixaMH.

§ 3. 03H8KH piBHOCTi TpHKyTHHKiB 45
B
MaJI. 61 MaJI. 62
10. IlepuMeTp pi:eH06eApeHoro TPHKYTHHKa AOPiBHI06 7,5 M, a 6i11-
Ha CTOpoHa AOPiBHIOG 2 M. 3HaiiAiTL OCHOBy.
11. IlepuMeTp piBHo6eApeHoro TPHKYTHHKa AOPiBHIOe 15,6 M.
3HaHAiTL HOI'O CTOPOHH, $1K~o: ·1) OCHOBa MeHIIla aa 6i11Hy
CTOpoHy Ha 3M; 2) OCHOBa 6iJILma BiA 6i'IHOl CTOpoHH Ha 3M.
12. AoseAiTL, ~o y piBHOCTOpOHHLOro 'l'PHKYTHHKa sci KYTH piBHi.
13. BiA sepmHHH C piBHo6eApeHoro TPHKYTHHKa ABC a OCHOBOIO
AB BiAKJiaAeHo piBHi BiApiaKH: CA1 - Ha CTopoHi CA i CB1 -
Ha cTopoui CB. AoseAiTL pisHiCTL TpHKYTHHKiB: 1) CAB1 i
CBA 1; 2) ABB, i BAA1.
14. Ha ocHoBi AB piBHo6eApeHoro TpHKYTHHKa ABC AaHo TO'IKH
A1 i B1. BiAOMO, ~o AB1 = BA1. AoBeAiTL, ~o TPHKYTHHKH
AB,C i BA1C piBHi.
15. TpHKYTHHKH ACC, i BCC1 piBHi. lx sepmHHH A i B JiemaTL
no piaHi 6oKH siA npHMo'i CC,. AoBeAiTh, ~o TPHKYTHHKH
ABC i ABC, piBH06eApeHi (MaJI. 62).
16. C«l>opMyJlIOHTe i AOBeAiTL TeOpeMy, o6epHeHY AO TBepAmeHH$1
aaAa'li 12.
17. Ha cTopouax AC i BC TpHKYTHHKa ABC BaHTo TO'IKH C 1 i C2.
AoeeAiTL, ~o TPHKYTHHK ABC pi:eH06eApeHHH, HK~o TPHKYT-
HHKH ABC1 i BAC2 piBHi (MaJI. 63).
18. 1) AoeeAiTL, ~o cepeAHHH cropiH pi:eHo6eApeuoro TPHKYTHHKa
e TaKom BepmHHaMH piBHOOeApeHoro TPHKYTHHKa.
2) AoeeAiTh, ~o cepeAHHH cropiH piBHocTOpoHHLoro TPHKYT-
HHKa e TaKom BepmHHaMH piBHOCTOpOHHLOrO TPHKYTHHKa.
19. 1) HaKpecJiiTL TPHKYTHHK a rOCTPHMH KYTaMH. 3a AODOMOrOIO
KOCHH~$1 i JiiHiHKH npoeeAiTh y HLOMY BHCOTH. BHKOHaii:Te
BnpaBy noBTOpHo AJIH TPHKYTHHKa, y HKoro OAHH KYT Tynuii:.

1 KJiac 46
[ E
A
Man. 63 MaJI. 64
2) HaKpec.ni.Ti. TPHKYTHHK. 3a AOnoMoroio TpaucnoPTupa i .rri-
uiiiKH npose,u;iTI> y HbOMY 6iceKTPHCH.
3) H&Kpec.ni.Ti. TPHKYTHHK. 3a AOnoMoroio niHiiiKH a noAiJIKa-
MH npose,u;iTb y HbOMY Mep;iaHH.
20• .D;oseAiTi., ~o y piBHo6eApeHoMy TPHKYTHHKy: 1) 6iceKTpHCH,
nposeAeHi a sepmRH npH ocuosi, piBHi; 2) Me,u;iauu, nposeAeHi
3 THX caMHX eepmHH, TaKOXC piBHi.
21. .D;oseAiTi., ~o y piBHHx TpHKYTHHKaX ABC i A 1B1C1: 1) MeAia-
HH, nposeAeHi a sepIQHH A i Ai, piBui; 2) 6iceKTpucu, npoBeAe-
ui a sepmHH A i Ai, pisui.
22. To11KH A, B, C, D nexcan ua OAHiii npBMiit, npH'IOMY sip;piaKH
AB i CD MaIOTh cniJii.uy cepeAHHY• .D;oseAiTh, ~o KOJIH TPHKYT-
HHK ABE piBHo6eApeHHii a ocuosoio AB, TO TpHKYTHHK CDE
a ocuosoio CD TaKoxc pisuo6e.u;peud (Man. 64).
23. .D;ose.u;iTb piBHiCTh TPHKYTHHKiB aa KYTOM, 6iceKTPHCOIO ~oro
Kyra i CTOpOHOIO, npunerJIOIO AO HbOI'O.
24. Y piBHo6eApeHOMY TPHKYTHHKY ABC a ocuoeoio AC npoBeAeHo
MeAiauy BM. Ha uiii yawro TO'IKY D. .D;oseAiTL pieHicTi. TpHKYT-
HHKis: 1) ABD i CBD; 2) AMD i CMD.
25. .D;oeeAiTi., ~o TPHKYTHHK ABC piBHo6eApeHHii, SK~o y ui.oro:
1) Me,u;iaua BD e BHCOTOIO; 2) BHCOTa BD e 6iceKTpuco10;
3) 6iceKTpuca BD e Mep;iauoio.
26. .D;auo ABa piBHo6eApeHHX TPHKYTHHKH ia CniJibHOIO OCHOBOIO.
.D;oseAiTi., ~o ix MeAiaHH, npoBeAeHi AO ocHOBH, JiemaTi. ua OA-
uiii npaMiii.
27. Y piBHo6eApeHOMY TPHKYTHHKY ABC a ocuoeo10 AC nposeAeHo
MeAiauy BD. 3uaiiAiTb IT AOBmuuy, aK~o nepuMeTP TPHKYTHH-
Ka ABC AOpisuioe 50 M, a TPHKYTHHKa ABD - 40 M.

§ 3. 03H8J(H piBHOCTi TpHICyrHHKiB 47
[
A D
A
Man. 65 Man. 66
28. ,lJ;oBeAin, JIO 6iceKTpHca piBHo6eApeHoro TPHKYTHHKa, npoBe-
AeHa 3 sepmHHH, npoTHJie:>KHOi OCHOBi, G MeAiaHOIO i BHCOTOIO.
29. Y TPHKYTHHKax ABC i A1B1C1: AB= A1B1, AC= A1C1,
LC= L Ci= 90°. ,lJ;oBeAiTi., JIO /:::.ABC= /:::. A1B1C1.
30. ,lJ;oBeA.iTb, IIO y piBHo6eApeHOMY TPHKYTHHKY BHCOTa, OBYJIeHa
Ha OCHOBY, e MeAiaHOIO i 6iceKTPHCOIO.
31. TpuKyTHHKH ABC i ABC1 i3 cnini.HoIO OCHOBOIO AB piBH0-
6eApeHi. ,lJ;oeeAiTi. piBHicTi. TpHKYTHHKiB ACC1 i BCC1.
32~ To'IKH A, B, C, D JiemaTi. Ha OAHiii npaMiii. ,lJ;oBeAiTi., JIO KOJIH
TpHKYTHHKH ABE1 i ABE2 piBHi, TO TPHKYTHHKH CDE1 i CDE2
TaKO:>K piBHi (MaJI. 65).
33. ,D;ea BiApi3KH AB i CD nepeTHHaIOTbcR y TO'l~ 0, &Ka e cepeAH-
HOIO KO:>KHoro 3 HHX. ,lJ;oBeAiTb piBHicTi. TPHKYTHHKiB ACD i
BDC.
34. ,lJ;oBeAiTb piBHiCTb TPHKYTHHKiB 3a ABOMa CTOpoHaMH i MeAia-
HOIO, npoBeAeHOIO AO OAffiG'i 3 HHX.
35. BiApi3Ku AB i CD nepeTHHa10nca. ,lJ;oBeAiTi., IIO KOJIH BiApi3KH
AC, CB, BD i AD piBHi, TO npoMiHi. AB e 6iceKTpHco10 KyTa
CAD i npoMiHi. CD - 6iceKTpHco10 KyTa ACB (MaJI. 66).
36~ ,D;oseAiTi., JIO B 3&Aa11i 35 npm.d AB i CD nepneftAHKYJI.RpHi.
37. TpHKyTHHKH ABC i BAD piBHi, npH'lOMY TO'IKH Ci D JiemaTb
no pi3Hi 6oKu BiA npaMo'i AB (MaJI. 67). ,lJ;oBeAiTi., IIO: 1) TPH-
KYTHHKH CBD i DAC piBHi; 2) npaMa CD AiJIHTb BiApiaoK
AB nonoJiaM.
38. BiApi3KH AB i CD OAHaKoBO'i AOBmHHH nepeTHHaIOTbCR B TO'I-
~ 0 TaK, JIO AO= OD. ,lJ;oBeAiTi. piBHicTb TPHKYTHHKiB ABC
i DCB.

7 KJiac 48
[
Mil.II. 67 Man. 68
39. ,IJ;oBeAiTb piBHiCTb TPHKYTHHKiB 38 ABOM8 CTOpoH8MH i MeAi8-
HOI<>, ~o BHXOA11Tb 3 O,ll;Hie'i BepmHHH (Ma.n. 68).
40. ,IJ;oBeAiTb piBHiCTb TPHKYTHHKiB 38 CTOpOHOI<>, MeJU8HOIO, npo-
BeAeHOI<> AO He'i, i KYT8MH, 11Ki YTBOPI<>e MeAi8H8 i3 CTOpoHOI<>.
§ 4. CYMA KYTIB TPHKYTBllKA
29. IlAPAJIEJILIDCTb IlP.HMHX
Teo p e M 8 4.1. /.fsi npllMi, na.pauJ1.1>Hi rperi4, napa.J&ea'l>Hi
ooHa o8Hiu.
,n; o Be A e H H 11. Hexaii npaMi a i b n8p8Jiem.Hi npaMiit c. IlpH-
nycTHMO, ~o np11Mi a i b He n8p8JieJibHi (M8JI. 69). ToAi BOHH nepe-
THYTbCS B ,u;eBKiii To~i C. OTme, 11epe3 TO'IKY C npoxo,u;&Tb ABi
npsMi, n8paJieJI1>Hi npaMiit c. AJie ~e HeMomJIHBO, 6o qepe3 roqKy,
~o He Jie)KHTb Ha ,ii;aHiii npaMiii, MO)KHa npoBecTH He 6iJibm SK OAHY
npaMy, napaJieJibHY A8Hiii. TeopeMy ,u;oBe,u;eHo.
3 a A a 11 a (4). Ilp11Mi AB i CD napaJieJII>Hi. ,IJ;oBeAiTI>, ~o
KOJIH si,ll;piooK BC nepeTHHae npsMy AD, TO ToqKa nepeTHHY
H8Jl0)KHTb Bi,ll;pi3Ky AD (M&.11. 70).
a
: [
II
x
c
Man. 69 Man. 70

§ 4. CyMa KyTiB TPHICYTHHKa 49
Po a B' SI a a H HH. Hexaii X - TO''IKa nepeTJrny Bi,z:i;piaKa
BC a npSIMOIO AD. IIpoBe,z:i;eMo 11epea Hei'. npnMy x, napaJieJibHY
npSIMiii AB. BoHa 6y,z:i;e napaJieJibHa i npSIMiii CD. IIpSIMa x
poa6HBae nJio~HY Ha ,z:i;Bi niBnJIOID;HHH. To'IKH Bi C JiemaTb
y piaHHX niBnJIOID;HHax, 60 Bi,z:i;piaoK BC nepeTHHae npSIMy x
(B TO'IW X). To11Ka A JiemHTb y Tiii caMiii niBnJIOID;HHi, m;o i B,
a TO'IKa D - y TiH caMili niBilJIO~Hi, m;o i c. ToMy Bi,z:i;pi-
30K AD nepeTHHae npSIMY x. A TO'IKOIO nepeTHHY e TO'IKa
X Bi,z:i;piaKa BC.
30. KYTH, YTBOPEHI B PE3YJibTATI IIEPETHHY
,ll.BOX IIPHMHX CI1IHOIO
Hexaii AB i CD - ,z:i;Bi npSIMi i AC - TpeTSI npnMa, m;o nepeTH-
Hae npnMi AB i CD (MaJI. 71). IIpSIMa AC Bi,z:i;HOCHO npHMHX AB i
CD HaaHBa6TbCH ci'lno10.
IIapH KyTiB, SIKi yTBOJ>IOIOTbCH B peayJibTaTi nepeTHHY npHMHX
AB i CD ci1JHOIO AC, MaIOTb cnewaJibHi Ha3BH.
Hl<m;o TO'IKH Bi D JiemaTb B o,z:i;Hiii niBnJIOID;HHi BiAHOCHO npaMoi'.
AC, TO KYTH BAC i DCA Ha3HBaIOTbCJI 8HyrpiwniMU oanocropon-
niMu (Man. 71, a).
Hl<m;o TO'IKH B i D nemaTb y piaHHX niBnno~Hax BiAHOCHO
npnMoi'. AC, TO KYTH BAC i DCA HaaHBaIOTbCH BHyrpiwniMu pi3no-
croponniMu (Man. 71, 6).
Ci11Ha AC yTBop10e a npaMHMH AB i CD ,z:i;Bi napH BHyTpimHix
OAHOCTOJ>OHHix i ,z:i;si napH BHyrpimHix piaHOCTOJ>OHHix KyTiB. BHyT-
8
a)
MaJI. 71

7 KJiac 50
pimHi piaHocTopoHHi KYTH o,u;Hie'i napH,
HanpHKJia,u; L 1 i L 2, e cyMimHHMH a
BHYTpimHiMH piaHocTopoHHiMH KyTaMH
iHmo'i napH L 3 i L 4 (MaJI. 72). ToMy,
ftKIIJ;O BHYTPimHi pi3HOCTOpOHHi KYTH
O,!l;Hie'i napH piBHi, TO BHyTpimHi piaHO-
CTOPOHHi KYTH iHmo'i napH TaKO)I( piBHi.
Ilapa BHyTpimHiX piaHOCTOpoHHiX Ky-
TiB, HanpHKJia,u; L 1 i L 2, i napa BHyTpim-
MaJI. 72 Hix O,!l;HOCTOpOHHix KYTiB, HanpHKJiap;
L 2 i L 3, MaIOTb o,u;HH KYT cniJIL-
HHii - L 2, a p;Ba iHmi KYTH cyMhKHi, L 1 i L 3.
ToMy, BKIIJ;o BHyTpimHi piaHocTopoHHi KYTH piBHi, TO cyMa
BHyTpimHix op;HocTopoHHix KYTiB ,u;opiBHI06180°. I HaBnaKH, BKIIJ;O
CYMa BHYTPimHix O,ll;HOCTOpoHHix KyriB ,u;opiBHI06 180°, TO BHyTpim-
Hi piaHoCTopoHHi KYTH piBHi.
31. 03HAKA IIAPAJIEJibHOCTI IIP.HMHX
Teo p e Ma 4.2 (oaHaKa napaJieJibHOCTi npBMHX). HICUfO
BHyTpiw.Hi piaHOCTOpOHHi ICYTU piBHi a6o cyMa BHyTpiw.HiX OOHO-
CTOpoHHiX JCyTiB iJopi8H10e 180°, TO np1ud napaMlJbHi•
.n; o Be A e H Ha. Hexaii npaMi a i b YTBOpIOIOTb ia citJHOIO
AB piBHi BHyTpimHi piaHOCTOpoHHi KYTH (MaJI. 73, a). IlpHnyCTHMO,
npaMi a i b He napaJieJILHi, a oTme, nepeTHHaIOTbCB B ,u;eaxiii TO'I-
~ C (MaJI. 73, b).
Ci•IHa AB poa6HBae nJIOIIJ;HHY Ha p;Bi niBnJIOIIJ;HHH. B op;Hiii a
HHX Jie»eHTb TO'IKa C. Ilo6y,u;yeMo TpHKYTHHK BAC1, w;o ,u;opiBHI06
a a
h
[ /
<
'
MaJI. 73

§ 4. CyMa KyTi~ TPHKYTHHKa 51
TPHKYTHHKy ABC, 3 sepwHHOIO C1 B µ;pyriii niBIIJIO~Hi. 3a
YMOBOIO BHyTpiwHi pi3HOCTOpOHHi KYTH npH napaJieJILHHX a, b i ci'q-
Hiii AB pisai. OcKiJILKH si,ZJ;nosiµ;Hi KYTH TPHKYTHHKiB ABC i BAC
3 sepwHHaMH A i B piBHi, TO BOHH 36iraIOTLC.H 3 BHyTpiwHiMH
pi3HocTopoHHiMH KyTaMH. OT)Ke, np.HMa AC1 36irasTLC.H 3
np.HMOIO a, a np.HMa BC1 36ira6TLC.H 3 np.HMOIO b. ,D;icTaHeMo, m;o
11epe3 TO'IKH C i C1 npoxo,ZJ;.HTL µ;si pi3Hi np.HMi a i b. A :a;e HeMo>K-
JIHBo. 0Tme, np.HMi a i b - napaJieJILHi•
.HKm;o npu nepeTHHi np.HMHX a i b ci11Ho10 AB CYMa BHyTpiwHix
O,ZJ;HOCTOpOHHix KyriB µ;opiBHI06 180°' TO, .HK MH 3Ha&MO, BHyTpiwHi
pi3HOCTOpoHHi KYTH piBHi. 0Tme, .HK 6yJio ,ZJ;OBe,ZJ;eHO BHm;e, np.HMi
a i b - napaJie.n:LHi. TeopeMy ,11;ose,11;eao.
3 TeopeMH 4.2 BHilJIHBae, m;o iJBi nps;1d, nepneKiJuxyaspKi iJo
rper1>o'i, napa11.e1i1>Ki•
.HKm;O y napi BHyTpimHix pi3HOCTOpOHHiX KyTiB O,ZJ;HH 3 KyTiB
3aMiHHTH BePTffKaJILHHM iioMy, TO µ;icTaHeMo napy KYTiB, .HKi Ha3H-
BaIOTLC.H BianoBianUMu ICYTaMU, yTsopeHHMH B pe3yJILTaTi nepeTu-
ay ,ZJ;BOX np.HMHX ci'IHOIO.
KyTH 1 i 2 aa MaJIIOHKY 74 - BHyTpimai pi3HocropoHHi, a KYTH
1 i 3 - Bi,ZJ;noBi,ZJ;Hi.
3 piBHOCTi BHyTpiwHiX pi3HOCTOpOHHix KyTiB BHilJIHBae piBHiCTL
Bi,ZJ;noBi,ZJ;HHX KyTiB i HaBnaKH. 3Bi,ZJ;CH µ;icTaHeMO 03HaKy napaJieJIL-
HOCTi np.HMHX 3a Bi,Zl;IloBi,ZJ;HHMH KyTaMH. A caMe: np1UCi riapa1ie1i1>-
Ki, BKUJO BiiJnoswKi Kyru piBKi.
t1 3 aµ; a 11 a (8). ,D;aao np.HMY AB i TO'IKY C, m;o He Jie>KHTL
v Ha :u;iii np.HMiH. ,ll;oBe,ZJ;iTL, m;o 11epe3 TO'IKY c MO)KHa npoBeCTH
npsMy, napaJieJILHY np.HMiH AB.
Po 3 s' .H 3 a H H .H. IIp.HMa AC po36usae IIJIOIIJ;HHY Ha µ;si
niBnJio~HH (MaJI. 75). To•IKa B Jie>KHTi. B oµ;aiii 3 HHX. Bi,ZJ;-
J 0
B
MaJI. 74 MaJI. 75

7 KJiac 52
KJia.zi;eMo ui.zi; niunpaMoi CA y .zi;pyry niumxo~HHY KYT ACD,
~o .zi;opiBHIOO KYTY CAB. To.zi;i npaMi AB i CD napaneJILHi.
Cnpau.zi;i, AJIR IVIX npRMHX i ci11Hoi AC KYTH BAC i DCA
BHYTPiIIlHi pi3HOCTOpoHHi. OcKinbKH BOHH piBHi, TO npaMi
AB i CD napanenLHi.
3icTauna1011H TBep.zi;meHHa aa.zi;a11i 8 ii aKcioMH IX (ocHOBHoi
BJI8CTHBOCTi napaneJlbHHX npaMHX), npHXO)l;HMO .zi;o uamnHBoro
BHCHOBKy: 1'epea T01'Xy, 140 He lU!.UTb Ha iJaHiu npJU&iu, M031tH.a
npo6ecru napa;ieJibHY iu np11.My i riJibKU oiJHy.
32. BJIACTHBICTb KYTJB,
YTBOPEHHX IIPB IIEPETHID
IIAPAJIEJILHHX IIPHMBX CPIHOIO
Teo p e Ma 4.3 (o6epHeHa .zi;o TeopeMH 4.2). Hx140 iJBi napa-
lU!JibHi npR.Mi neperHyTi TpeTbOIO npR.M010, TO 6HyTpiWHi piaHOCTO-
poHHi xyTU pi6Hi, a cyMa BHyTpiutHix oOHOCTOpoHHix xyTi6
iJopi6H10e 180°.
,ZJ; o u e .zi; e H Ha. Hexaii a i b - napanenLHi npaMi i c - npa-
Ma, aKa nepeTHHae ix y TO'IKax A i B. IIpoue.zi;eMo 11epea TO'IKY A
npaMy aI TaK, ~06 BffYTpimHi piaHOCTOpoHHi KYTH, YTBOpeHi
ci'IHOIO c i npaMHMH a. i b, 6ynH piBHi (Man. 76).
3a oaHaKoIO napanenLHOCTi npaMHX npaMi a1 i b napanenLHi.
OcKinLKH 11epea TO'IKY A npoxo.zi;HTL TiJlbKH o.zi;Ha npaMa, napaJieJIL-
Ha npaMiii b, TO npaMa a a6iraeTLCa a npRMoIO a 1• 0Tme, BHyTpimHi
piaHocTopOHHi KYTH, yTuopeHi ci11Ho10 ii napanen1>HHMH npaMHMH
a i b, piBHi. TeopeMy .zi;oue.zi;eao.
3 BJiaCTHBOCTi KyTiB, YTBOpeHHX npH nepeTHHi napaJienbHHX
npaMHX Ci'IHOIO, BHUJIHB86, ~o XOJiU
--=.,..,.--=~--- a,
a
Mu. 76
nplUUI nepneHiJu«y;i11.pHa iJo oiJHie'i
a napalU!JibHUX npR.MUX, TO 601'4 nep-
neHiJuKy1'R.pHa i iJo iJpyzoi.
3 a A a 11 a (13). IIpaMi AC
i BD napaJieJI1>Hi, npH'IOMY
TO'IKH Ai D nemaT1> no piaHi
6oKH Bi.A ci11Hoi BC (Man. 77).
,ZJ;oue.zi;iT1>, ~o: 1) KYTH DBC
iACBuffY'l'PllmdpiaHocTopoHHi
ui.zi;Hocao ci11Hoi BC; 2) npo~

§ 4. CyMa KyTiB TpHKyTHHKa
BC Tipoxo,n;HTh Mim CTopoHaMH KyTa
ABD; 3) KYTH CAB i DBA BHyTpimHi
O,ll;HOCTOpOHHi Bi,ll;HOCHO Ci'IHOl AB.
PoaB'aaaHHa. 1) KyTH DBC
i ACB BHyTpimHi piaHOCTopoHHi TOMy,
m;o TO'IKH A i D JiemaTh TIO piaHi 6oKH
BiA ci11Hoi BC.
2) IlpoMiHh BC Tipoxo,n;HTh Mim
cTopoHaMH KyTa ABD TOMY, m;o BiH
TiepeTHHae Bi,n;piaoK AD a KiHD;HMH Ha MaJI. 77
CTOpoHaX KyTa (aa,n;a11a 4).
53
3) KyTH CAB i DBA BHYTPimHi o,n;HocTopoHHi TOMY,
m;o TO'IKH C i D JiemaTh TIO o,n;HH 6iK ei,n; ci11Hoi AB, To6To
y niBTIJIOI.lHHi, ,n;e Jie)KHTb TO'IKa x TiepeTHHy Bi,n;piaKiB
BC i AD.
33. CYMA KYTIB TPHKYTHHKA
Teo p e M a 4.4. CyMa KyTiB TpuKyTnuKa iJopisn10e 180°.
,II; o Be A e H H a. Hexa:H ABC - ,n;aHHH TPHKYTHHK. IlpoBe,n;eMo
11epea eepmHHy B TipHMy, TiapaJieJI1>Hy TipHMiii AC. BiabMeMo Ha Hiii
To11Ky D TaK, m;o6 TO'IKH A i D JiemaJIH TIO piaHi 6oKH ei,n; TipH-
MOi BC (MaJI. 78).
KyTH DBC i ACB piBHi HK BHYTpimHi piaHOCTOpoHHi TIPH Tiepe-
THHi TiapaJieJibHHX TIPHMHX AC i BD ci11HOIO BC. ToMy cyMa KYTiB
TPHKYTHHKa TIPH BepmHHax Bi C ,n;opiBHIOe KYTY ABD.
A cyMa Bcix Tpbox KYTiB TPHKYTHHKa ,n;opiBHIOe cyMi KYTiB ABD
i BAC. OcKiJibKH u;i KYTH BHYTPimHi o,n;HocropoHHi AJIH TiapaJieJib-
HHX AC i BD i ci11Hoi AB, TO ix cyMa ,n;opiBHIOe 180°. TeopeMy
AOBep;eHO.
3 TOOpeMH 4.4 BHTIJIHBae, m;o y 6yiJ1,-
1'1WMY TPfU'YTHU"Y npuHallMHi iJsa 1'1JTU
ZOCTpi.
CTipaB,n;i, TIPHTIYCTHMO, m;o y TPHKYT-
HHKY TiJibKH O,ll;HH rocTpHH KYT a6o B3a-
raJii HeMae rocTpHx KYTiB. To,n;i y u;wMy
TPHKYTHHKY e p;Ba KYTH, KomHHH a
$1KHX He MeHmHii aa 90°. CyMa u;Hx ,n;Box
KYTiB yme He MeHma aa 180°.
A
8 D
[
MaJI. 78

7 KJiaC 54
A.n:e ~e HeMO>KJIHBO, 60 cyMa Bcix KYTiB TPHKYTHHKa p;opiBHI06
180°. ID;o ii Tpe6a 6yJIO ,ll;OBeCTH.
t1 3 a p; a 11 a 30. 1loMy p;opiBHIOIOTb KYTH piBHOCTOpOHHi.oro
v TpHKYTHHKa?
p 0 a B' Ha a H H H. y piBHOCTOpOHHbOMY TPHKYTHHKY' HK
Bi,ll;OMO, BCi KYTH piBHi. OcKiJibKH BOHH B cyMi CTaHOBJIHTb
180°, TO KO>KHHH 3 HHX p;opiBHI06 60°.
34. 30BHIWHI KYTH TPHKYTHHKA
8oeniwni.M KyroM TpHKYTHHKa npH p;aHi:H sepmHHi H83HB86-
TbCH KYT' cyMi>KHHH a KYTOM TPHKYTHHKa npH aj.:H BepmHHi
(MaJI. 79).
ID;o6 He nJiyTaTH KYT TpHKYTHHKa npH p;aHi:H BepmHHi ia aosHim-
HiM KYTOM npH aj.ii caMiH BepmHHi, :Horo iHop;i Ha3HBaIOTb 6Hyr-
piwniM KyroM.
Te o p e M a 4.5. 3osHimH.iu "YT Tpu1CyTHu1Ca iJopisH10e cyJtd
iJsox BHyTpimHiX JCyTiB, He cyMiJfeHUX a HUM.
Ao B e p; e H H s. Hexa:H ABC - p;aHHii TpHKYTHHK (MaJI. 80).
3a TeopeMoIO 4.4 L A + L B + L C = 180°. 3Bip;cH BHnJIHBa6,
~o L A + L B = 180° - L C. Y npaBi:H 11acTHHi aj.el piBHocTi
CTOITb rpa.u;ycHa Mipa aoBHimHi.oro KyTa TPHKYTHHKa npH BepmH-
Hi C. TeopeMy .u;oue,u;eHo.
3 TeopeMH 4.5 BHilJIHBae, ~o aosHimH.iu "YT TpBlCYTHUJCa 6in.'b-
UJ,UU · siiJ 6yiJ'b-lf.1COZO BHyTpiUJ,H'bOZO JCyTa, He cyMiJfeHOZO a HUM.
3 a .a; a 11 a 35. Y TpHKYTHHKY ABC npoBep;eHo BHCOTY
CD. SlKa a Tpi.ox TO'IOK A, B, D Jie>KHTb Mim ,u;BoMa iHmHMH,
HK~o KYTH A i B TpHKYTHHKa rocTpi?
Po as' H·a a H H s. To11Ka B He Mome JiemaTH Mi>K TO'IKa-
MaJI. 79

§ 4. CyMa KyTiB TpHKYTHHK8 55
A [ D B A
Man. 80 Man. 81
MH Ai D. HK6H BOHa 6yna Mim ToqKaMH Ai D (Man. 81), TO
rocTpHii KYT ABC HK 30BHimHiii KYT TpHKYTHHKa CBD 6yB 6H
6inLmHM BiA npaMoro KyTa CDB. TaK caMo AOBOAHMO, ~o ii
T011Ka A He Mome nemaTH Mim TO'IKaMH Bi D. Orme, TO'lKa D
nemHTL Mim TO'IKaMH A i B.
35. IlPSIMOKYTHHH TPHKYTHHK
TpHKYTHHK Ha3HBa6TLca npaMmcyrHuM, aK~o BiH Mas np.RMHii
KyT.
OcKinLKH cyMa KyTill TpHKYTHHKa AOPiBHI06 180°, TO B npaMo-
KYTHOMY TPHKYTHHKY TinLKH OAHH np8MHH KyT. ,ll;Ba iHmHX KYTH
npaMOKYTHoro TpHKYTHHKa rocTpi. CyMa rocTpHx KyTiB npaMoKyT-
Horo TPHKYTHHKa AOPiBHI06 180° - 90° = 90°.
CropoHa npaMOKYTHoro TPHKYTHHKa, ~o nemHTL npoTH npaMo-
ro KyTa, Ha3HBaeTLC.R zinoreHy3010, p;Bi iHmi CTOpoHH Ha3HBaIOTbC.R
KareraMu (Man. 82).
81 B
~
~
~
t:::i
<::
Kamem A, c, A [
Man. 82 Man. 83

7 KJIBC
0
8
[ A
MaJI. 84
KyTOM 30° KaTeT'
56
BiA3Ha'IHMO TaKy oaHaKy pisHocTi
np$1MOKYTHHX TpHKYTHHKiB aa rinoTeHy-
3010 i KaTeTOM:
HK140 zinoreHy3a i KaTeT oiJHoto
npIU&OKYTHOtO Tpu.KyrHUKa triOnoBWHO
iJOpUmlOHYn tinmeHyai i KaTeTg iJpytoto
rpuKyTHUKa, TO TaKi TpUKgTHUKU pi8Hi
(MaJI. 83).
,ZJ;oBe,D;eHHR W6l 03HRKH ARHO Y BH-
rJIR,ll;i poou'RaaHHR aa,11;a11i 29 a § 3.
rf 3 a A a ti a (43). ,ZJ;oue,o;iTL, ~o
V B npSIMOKYTHOMY TPHKYTHHKY 3
npoTHJie>KHHA U,bOMY KyTY' ,11;opiBHJOe
llOJIOBHHi rinoTeHy3H.
Po a u' SI a a H HR. Hexaii ABC - npRMOKYTHHii TPHKYT-
HHK 3 npSIMHM KYTOM c i KYTOM B, ~o ,o;opiBHIOG 30° (MaJI. 84).
Ilo6y,o;yeMo TPHKYTHHK DBC, ~o ,o;opiBHIOG TpHKYTHHKY ABC,
S1K noKaaaHo Ha MaJIIOHKY 84. Y TPHKYTHHKy ABD Bci KYTH
pisai i ,o;opiBHIOIOTL no 60°, TOMY siH pisaocTopoHHiii. OcKinL-
1 1 ~
KH AC= -yAD, a AD= AB, TO AC= -yAB, ~o H Tpe6a
6yJIO ,ll;OBecTH.
36. ICllYBAHHSI I GAHIDCTb
IIEPllEHAHKYJUIPA lJ.O IIP.HMOI
Teo p e Ma 4.6. 3 6yiJ•-11KOi T01'KU, 140 He .11e.un "" ilaHiu
np11Mi4, M.031&Ha onycruru "" lflO npIU&y nepneHiJUKY""P i ri1'•1'u
oiJuH.
,ZJ; o Be ,11; e H HSI. Hexaii a - ,o;aHa npRMa ii A - TO'IKa, ~o
He JiemHTh Ha Hiii (MaJI. 85). Ilpose,o;eMO 11epe3 ,ll;OBiJILHY TO'IKY
UpSIMOl a nepneH,D;HKYJISIPHY npSIMy. Ilpose,o;eMO Tenep 11epea TO'IKY
A napaJieJihHY iii npRMY b. Boua 6yAe nepneH,D;HKYJIRpHOIO AO npH-
Mo'i a, OCKi.nbKH npa11a a, 6y,IU"IH nepneHAHKYJIRpHOIO AO OAHi6i 3
napaJieJI&HHX npRMHX, nepneHAHKYJIHpua i AO APYroi. Bi.ApiaoK AB
IlpHMOi bi e nepneHAHKYJIRp, nposeAeHHH a TO'IKH A AO npRMoi a.
,ZJ;oeeAeM.o 6.D;HHiCT& nepneHAHKYJI.Hpa AB. IlpBIIycTRMO icHy-
BaHH.H iumoro nepneHAHKynapa AC. To,ll;i TPHKYTHHK ABC MaTHMe
ABa upaMHX KYTH. A u,e, HK si.,ll;oMo, ueMmKJIHBO. TeopeMy .u;oseAeHo.

§ 4. CyMa xyTiB TPHKYTHHKa 57
b
A
a
B a b
81 B
Man. 85 Man. 86
,IJ;oBJKHHa nepneap;HKyJisq>a, on~eaoro 3 p;aaoi TO'IKH Ha npHMy,
H83HB86Tl.CH 8WCTaHH10 8W TO'llCU 80 npA.Moi.
3 a A a 11 a (50)•..,IJ;oBep;iTh, m;o ~ip;cTaHi.. B~ .6yp;L-HKHX
ABOX TO'IOK npaMOl AO napaJieJihHOl npHMOl plBHl.
Po 3 B' H 3 a a a H. Hexaii a i b - napaJieJihHi npHMi i A,
A 1- p;oBiJihHi TO'IKH Ha npHMiii a (MaJI. 86). OnycTHMO 3 TO'l-
KH A1 nepneap;HKYJIHI> A1B1 Ha npaMy b. Bi,u;KJiap;eMo Dip;
TO'IKH B1 Ha npodii b Bi.n;piaoK B1B, m;o p;opiBHIOO Bi,u;piaKy
AA 1 TaK, m;o6 TO'IKH A 1 i B 6yJIH no piaai OOKH Bi.n; npHMoi
AB1. Top;i TpHKYTHHKH AB1A1 i B1AB piBHi 3a nepmo10 03-
HaK010. Y HHX CTOpoHa AB1 cniJILHa, AA1 = BB1 3a no6yp;o-
BOIO, a KYTH B1AA1 i AB1B piBHi HK BHyTpimHi piaHocropoa-
Hi npH nepeTHHi napaJieJILHHX a Ta b 3 ci11Ho10 AB1. 3 piBHOCTi
TPHKYTHHKiB BHilJIHBae, m;o AB - nepneap;HKYJIHI> p;o npHMoi
b i AB= A1Bi. m;o ii Tpe6a 6yJio p;oBecTH.
HK 6a'IHMO, Bi.n;cTaHi Bi.n; ycix ro110K npHMoi .n;o napaJieJILHoi
npHMoi piBHi. ToMy KaJKYTh, m;o napaJieJILHi npHMi piBHOBi,u;p;aJieHi.
BiiJcTann10 Mi31C napa.n.e.n.bHUMU npsutuMu Ha3HB86TLCH Bi.n;-
cTaHL Bip; HKOi-ae6yp;L TO'IKH OAHiei npHMOi AO p;pyroi npHMOi.
37. 3 ICTOPII BHHHKHEHIDI rEOMETPII
Ilepmi Bi,u;OMOCTi npo BJiaCTHBOCTi reoMeTpH'IHHX cl>iryp JIIO,D;H
.n;iCTaJIH B peayJILTaTi npaKTH'IHOl p;iaJihHOCTi i CDOCTepeJKeHb Hap;
HaBKOJIHWHiM CBiTOM. 3 11acoM y11eai noMiTHJIH, m;o p;eHKi BJiaCTH-
BOCTi reoMeTpH'IHHX cl>iryp MOJKHa BHBecTH 3 iHWHX BJiaCTHBOCTeii
MipKyBaHHHMH. TaK BHHBKJIH TeopeMH i p;osep;eHHH.
3'HBHJIOCJI npHpop;He 6amaliHH no MOJKJIHBOCTi 3MeHWHTH KiJIL-
KiCTL THX BJiaCTHBOCTeii reoMeTPH'IHHX cl>iryp, HKi BHDJIHBalOTh
6e3nocepep;HLO 3 ,u;ocBip;y. TBepp;meHHa, ·m;o 3aJIHWHJIHCH 6e3

7 KJI8C 58
,ZJ;OBe,ZJ;eHHR BJiaCTHBOCTeH, CTaJIH aKcioMa-
MH. TaKHM qHHOM, aKcioMH MaIOTb ,ZJ;ocei,ZJ;-
He noxo,ZJ;meHHa.
reoMeTpiR Ha paHHiH CTa,ZJ;fi CBOrO p03-
BHTKY ,ZJ;OCarJia oco6JIHBO BHCOKOrO piBHR
e SranTi. Y nepmoMy THCSAOJii'M'i AO Hamoi
epH reOMeTpH'IHi ei,ZJ;OMOCTi Bi,ZJ; erHilTRH
nepeihuJIH AO rpeKie. 3a nepioA a VII no
III CTOJiiTTs AO Hamoi epH rpe~i reo-
MeTpH He JIHme a6araTHJIH reoMeTpi10
qHcJieHHHMH HOBHMH TeOpeMaMH, aJie
M. I. Jio6a"tJ.escLKHii - apo6HJIH TaKOm cepiioaHi KpoKH AO cTpororo
pociii:cLxd MaTeMaTHK li o6rpyHTyeaHHa. BaraToeiKoea po6oTa
(1792-1856) rpe~KHX reoMeTpie aa u;eii nepioA 6yJia
yaaraJibHeHa EBKJiiAOM (330-275 pp. AO
H. e.) e iioro aHaMeHHTiii npa~ cHaqaJia•.
BHKJiaA reoMeTpii e cHaqaJiax• EBKJii,ZJ;a no6y,ZJ;oeaHo Ha cHCTe-
Mi aKcioM. IJ;a CHCTeMa ei,ZJ;pi3HR6TbCR ei,ZJ; CHCTeMH aKcioM, npHHHR-
TOl y ,ZJ;aHOMY ni,ZJ;py'IHHKy. AJr.e B Hiii Tem e aKCiOMa napaJieJibHHX.
AKcioMa napaJieJibHHX, Ha ei,ZJ;MiHy ei,ZJ; iHmHX aKcioM, He Mae
HaoqHoro niATBepAmeHHa. MomJIHBO qepea u;e a qacie EBKJii,ZJ;a MaTe-
MaTHKH 6araT1>ox KpaiH HaMaraJIHCR AOBecTH ii HK TeopeMy. Ane u;e
HiKoMy He BA&BaJiocs. HapemTi, e XIX cToJiiTTi 6yJio ,ZJ;oee,ZJ;eHo,
m;o u;e apo6HTH HeMO>KJIHBO. IlepmHM, XTO BHCJIOBHB u;e TBepAmeH-
Ha, 6ye eeJIHKHii pociiicbKHii M&TeM&THK MHKoJia leaHOeHq JI06a-
qeec1>KHii.
? KOHTPOJILHISAITHTAHH.H
1. ,IJ;oee,ZJ;iTb, m;o ABi npaMi, napaJieJI1>Hi TpeTiii, napaJieJI1>Hi OAHa
O,ZJ;HiH.
2. IloscHiTb, aKi KYTH H&3HBaIOTbCR BHyTpimHiMH OAHOCTOpoHHi-
MH. HKi KYTH HaaHBalOTbca BHYTpimHiMH piaHOCTOpoHHiMH?
3. ,IJ;oee,ZJ;iTb, m;o KOJIH BHYTPimHi pi3HOCTOpoHHi KYTH O,ZJ;Hiei napH
pieHi, TO BHyTpimHi piaHOCTOpoHHi KYTH Apyroi napu TaKom
pieHi, a cyMa BHYTpimHix OAHOCTOpoHHix KYTiB KOmHOl napH
,ZJ;OpiBHIOG 180°.
4. ,IJ;oee,ZJ;iTb 03H&KY napaJieJibHOCTi npRMHX.
5. IloscHiTb, RKi KYTH H&3HBaIOTbCR ei,ZJ;noei,ZJ;HHMH. ,IJ;oee,ZJ;iTb, m;o

§ 4. CyMa KYTie TPHKYTHHKa 59
KOJIH BHyTpimHi pi3HOCTOpOHHi KYTH piBHi, TO Bip;noBip;Hi KYTH
TaKOm piBHi i HaBnaKH.
6. ,ll;oBep;iTb, ~o 11epea To11Ky, JIKa He nemHTb Ha p;a'Hiii rrpHMiii,
MOmHa npoBeCTH napaJieJibHY iH npHMy. CKiJibKH npHMHX,·
napaJieJibHHX p;aHiii, MomHa npoBecTH 11epea To11Ky, ~o He Jie-
mHTb Ha ~iii npsMiii?
7. ,ll;oBep;iTb, ~o KOJIH ABi napaJieJibHi npHMi nepeTHHaIOTbCS Tpe-
TbOIO npsMOIO, TO BHyTpimHi pi3HOCTOpOHHi KYTH piBHi, a cyMa
BHyTpimHix O,ll;HOCTOp<>HHiX KyTiB AOpiBHI06 180°.
8. ,ll;oBeAiTb, ~o ABi npHMi, nepneHAHKynspHi AO TpeTb01, napa-
JieJII>Hi. HK~o npHMa nepneH,IJ;HKyJispHa AO op;Hiei a p;Box napa-
JieJibHHX npHMHX, TO BOHa nepneHAHKYJIBpHa ii AO Apyrol.
9. ,ll;oBep;iTh, ~o cyMa KyTiB TPHKYTHHKa p;opiBHIOe 180°.
10. ,ll;oBep;iTb, ~o B 6yp;b-JIKOMY TpHKYTHHKY IIPHHaHMHi p;Ba KYTH
rocTpi.
11. ~o TaKe 30BHimHi:H KYT TpHKYTHHKa?
12. ,ll;oBeAiTb, ~o 30BHimHiH Kyt TPHKYTHHKa AOPiBHI06 cyMi ABOX
BHyTpimHix KyTiB, He cyMimHHX 3 HHM.
13. ,IJ;oBep;iTb, ~o aoBHimHiii KYT TPHKYTffHKa 6iJibmHii aa 6yp;b-
JIKHH BHyTpimHiii KYT' He cyMimHHii 3 HHM.
14. HKHii TPHKYTHHK Ha3HBa6TbCJI IIpJIMOKYTHHM?
15. lloMy AOPiBHIOe cyMa rocTpHx KyTiB rrpHMOKYTHoro TPHKYT-
HHKa?
16. HKa ia CTopiH npHMOKYTHOro TpHKYTHHKa HaaHBaeThcs: rin0Te-
uyao10? HKi cTopoHH HaaHBaIOTbCJI KaTeTaMH?
17. C4><>PMYJIIOHTe 03HaKy piBHOCTi npHMOKYTHHX TPHKYTHHKiB aa
rirrOTeHY30IO i KaTeTOM.
18. ,IJ;oBep;iTb, ~o 3 6y,11;b-JIKOl TO'IKH, mca He JiemHTb Ha p;aHiH
npHMiii, MOmHa onyCTHTH Ha ~IO npHMY nepneH,IJ;HKYJIS:p i TiJib-
KH OAHH.
19. ~o Ha3HBa6TbCS: Bip;cTaHHIO TO'IKH Bi,11; npJIMOi:?
20. IlOJICHiTb, ~o TaKe eip;CT8Hb Mim napaJieJibHHMH nps:MHMH•
• 3AAA'II
1. ,ll;oBep;iTb, ~o KOJIH Aes:Ka nps:Ma nepeTHHae OAHY 3 ABOX napa-
JieJibHHX npHMHX, TO BOHa nepeTHHae :H p;pyry.
2. ,IJ;oBep;iTb, ~o KOJIH ABi npHMi nepeTHHaIOTbCJI, TO 6yp;b-JIK8
TpeTs: npHMa nepeTHHae npHHaHMHi op;uy a ~x npHMHX.

7 KJiaC 60
3. .D;aHo a II b II c II d. .D;osep;iTL, ~o a II d.
4. IlpaMi AB i CD napaJieJILHi• .D;osep;iTL, ~o KOJIH uip;piaoK BC
nepeTHHae npaMy AD, TO TO'IKa nepeTHHY HaJie:>KHTL ui,ZJ;piaKy
AD (MaJI. 70).
5. .D;aHo TPHKYTHHK ABC. Ha CTOpoHi AB noaHa'leHo TO'IKY B 1,
a Ha CTopoHi AC - TO'IKY C1 (MaJI. 87). HaauiTL BHyTpimHi
OAHOCTOpoHHi i BHYTPimHi piaHocTopoHHi KYTH npH npSIMHX
AB, AC i ci'IHiii B1C1.
6. HaauiTL BHYTPimHi piaaocropoHHi i BHyTpimHi op;HocTopoHHi
KYTH Ha MaJIIOHKY 72.
7. Bi,ZJ;piaKH AD i BC nepeTHHaIOTLCH. ,IJ;JIH npHMHX AC i BD i ciq-
Hol BC HaauiTL napy BHYTPimHix piaHocropoHHix KYTiB. ,Il;JIH
n;ux me npSIMHX i ci'IHOI AB HaaBiTL napy BHYTPimHix op;Ho-
CTOpoHHix KyTiB. IlOHCHiTL Bi,ZJ;nOBi.,ZJ;b.
8. ,Il;aHo npaMy AB i TO'IKY C, SIKa He Jie:>KHTL Ha n;iii npaMiii.
.D;osep;iTL, ~o qepea TO'IKY C MO:>KHa npouecTH npHMy, napa-
JieJILHy npHMiii AB.
9. .D;ouep;iTL, ~o 6iceKTpHCH BHYTPimHix piaHOCTOpOHHix KyTiB,
yTBOpeHHX napaJieJILHHMH i ci'IHOIO, napaJieJILHi, T06To JiemaTL
Ha napaJieJILHHX npSIMHX.
10. Bi,ZJ;piaKH AB i CD nepeTHHaIOTLCH B roqn;i E i p;iJIHTLCH n;ie10
TO'IKOIO nonoJiaM• .D;oue,11;iTL, ~o npaMi AC i BD napaJieJILHi.
11. TpHKYTHHKH ABC i BAD piBHi. To'IKH Ci D Jie:>KaTL no piaHi
6oKH uip; npSIMol AB• .D;oue,11;iTL, ~o npaMi AC i BD napaJieJILHi.
12. KyT ABC ,11;opiBH10e 80°, a KYT BCD - 120°. 1IH MO:>KyTL npaMi
AB i CD 6yTH napaJieJILHHMH? 06rpyHTyiire ui.,ZJ;noBiAJ>.
13. IIpaMi AC i BD napaJieJILHi, npH'IOMy TO'IKH Ai D Jie:>KaTL no
piaHi 6oKH ui,ZJ; ci'IHOI BC (MaJI. 77). .D;oue,ZJ;iTL, ~o: 1) KYTH
DBC i ACB BHYTPimHi piaHOCTOPOHHi Bi.,ZJ;HOCHO Ci'IHOI BC;
2) npoMillL BC npoxop;HTL Mim cropoHaMH KyTa ABD; 3) KyTH
CAB i DBA BHYTpimHi O,ZJ;HOCTOPOHHi Bi,ZJ;HOCHO Ci'IHOI AB.
14. 1) PiaHH~ p;uox BHYTpimHix O,ZJ;HOCTOpoHHix KYTiB npH p;uox
napaJieJILHHX npSIMHX i Ci'IHiH p;opiBHI06 30°. 3Haii,ll;iTL n;i KYTH.
2) CyMa ,11;uox BHYTpimHix piaHocTOpoHHix KyTiB npH p;uox
napaJieJILHHX npHMHX i Ci'IHiii p;opiBHIOe 150°. 1IoMy p;opiB-
HIOIOTb n;i KyTH?
15. Op;HH a KYTiB, md yTBOpIOIOTLCH npu nepeTHHi p;uox napaJieJIL-
HHX npSIMHX ci'IHOIO, p;opiBHI06 72°. 3Haiip;iTL iHmi ciM KyTiB.
16. Op;HH a KyTiB, md yTBopIOIOTLCH npu nepeTHHi p;uox napaJieJIL-

§ 4. CyMa KYTIB TPHKYTHHKa 61
HHX npHMHX Ci'IHOIO, p;opiBHIOG 30°. ~ Mome OAHH 3 pemTH
ceMH KyTiB p;opiBHIOBaTH 70°? IloscHiTL B~IlOBi,D;L.
17. ,ll;oBep;iTL, m;o p;si npsMi, napaJie.nLHi nepnelfAHKYJIHPHHM nps-
MHM, nepneHAHKYJIHpHi Mim co6o10.
18. 3aaiip;iTL Hes~oMHii KYT TpHKYTHHKa, HKm;o 8 HLOMY p;sa KYTH
p;opiBHIOIOTL! 1) 50° i 30° ; 2) 40° i 75° ; 3) 65° i 80° ; 4) 25° i
120°.
19. 3Haiip;iTL KYTH TPHKYTHHKa, HKID;O BOHH nponop:o;iHHi AO 'IH-
ceJI: 1) 1, 2, 3; 2) 2, 3, 4; 3) 3, 4, 5; 4) 4, 5, 6; 5) 5, 6, 7.
20. ~ Mome y TPHKYTHHKY.OyTH! 1) ,D;Ba TYilHX KYTH; 2) TynHii i
npHMHH KYTH; 3) p;Ba npHMHX KyTH?
21. 1IH Mome 6yTH TynHM KYT npH OCHOBi piBHo6ep;peHoro TPHKyT-
HHKa?
22. 3aaiiAiTL KYT Mim 6i'IHHMH CTOpoHaMH piBHo6ep;peHoro TpH-
KYTHHKa, HKID;O KYT npH iioro OCHOBi p;opiBHIOG: 1) 40°; 2) 55° ;
3) 72°.
23. 3aaii,D;iTL KYT npH OCHOBi piBHo6ep;peHoro TPHKYTHHKa, BKID;O
KYT Mi>K 6i'IHHMH CTOpoHaMH p;opiBHIOe: 1) 80° ; 2) 120° ;
3) 30°. ,,
24. Op;HH a KyTiB piBHo6ep;peHoro TPHKYTHHKa p;opisHIOe 100°.
3Haiip;iTL pemTy KYTiB TPHKYTHHKa.
25. Op;HH a KyTiB pisao6ep;peaoro TPHKYTHHKa p;opiBHI06 70°.
3aaii,D;iTL pemTy KYTiB TPHKYTffHK&. CKiJILKH poaB'saKiB Mae
aap;aqa?
26. ,ll;oBep;iTL, m;o KOJIH O,D;HH 3 KYTiB piBHo6ep;peaoro TpHKYTHHKa
p;opiBHI06 60°, TO TpHKyTHHK - piBHOCTOpoHHiii.
27. YpiBHo6ep;peHOMY TPHKYTHHKY ABC 3 OCHOBOIO AC nposep;eHo
A [
Man. 87 Man. 88

7 KJiac 62
6iceKTJ>HCY CD. 3HaiiJtiTi. KYTH TJ>HKYTHHKa ABC, RKm;o KYT
ADC p;opiBHIOe: 1) 60°; 2) 75°; 3) a..
28. y piBHOOe,D;peHOMY TPHKYTHHKY ABC 3 OCHOBOIO AC i KYTOM
npH sepmmti B, m;o p;opiBHIOe 36°, nposep;eao 6icel(TJ>HCY AD.
,D;ose,ltiTb, m;o TJ>HKYTHHKH CDA i ADB piBHo6ep;peHi (Man. 88).
29. Y TpHKYTHHKY ABC nposep;eHo 6iceKTPHCH a sepmHH A i B.
To11Ky ix nepeTHHY noaHaqeHo D. 3HaiiJtiTi. KYT ADB, BKm;o:
1) LA= 50°, LB= 100°; 2) LA= a, L B = ~;
3) LC= 130°; 4) LC= y.
30. 'IoMy p;opiBHIOIOTb KYTH piBHOCTOpoHHbOro TpHKYTHHKa?
31. ~ JIKHM KYTOM nepeTHHaIOTbCJI 6iceKTPHCH ABOX BHYTpimHix
op;HocTopoHHix KYTiB npu napaJieJibHHX npsmtnx?
32. Op;ua ia aosHimHix KyTiB piBHo6ep;peHoro TPHKYTHHKa p;opis-
HI06 70°. 3Haiip;iTb KYTH TJ>HKYTHHKa.
33. 3uaiip;iTb KYTH TPHKYTHHKa, 3H8IO'IH, m;o 30BHimHi KYTH npH
,ll;BOX iioro BepJUHHaX p;opiBHIOIOTb 120° i 150°.
34. Asa aosuimHix KYTH TJ>HKYTHHKa p;opiBHIOIOTb 100° i 150°.
3uaii,ltiTb TJ>eTiii 30BHimuiA KyT.
35. Y TPHKYTHHKY ABC nposep;euo BHCOTy CD. HKa a TPbOX ToqoK
A, B, D neacHTb Miac ABOMa iumuMH, BKID;O KYTH A i B TJ>HKYT-
HHKa l'OCTpi?
36. Y TPHKYTHHKY ABC nposep;euo BHCOTY CD. HKa a Tpi.ox roqoK
A, B, D nemHTb Miac ABOMa iHmHMH, BKID;O KYT A TynHii?
0614pyHTyHTe Bip;noBi,ll;b.
37. Aosep;iTi., m;o 6iceKTpHca aosuimai.oro KyTa npH sepmHHi
piBHo6ep;peuoro TJ>HKYTHHKa napaJieJlbHa OCHOBi.
38. CyMa aosaimHix KYTiB TJ>HKYTHHKa ABC npu sepmHRax A i B,
BMTHX no O,ll;HOMy AJIJI KOXCHOl BepmHHH, p;opiBHI06 240°.
'IoMy p;opiBHIOe KYT C TPHKYTHHKa?
B
0 A [ E
Man. 89

§ 5. reoMeTpH'IHi IIOOYAOBH 63
39. ,D;aHo TpHKYTHHK ABC. Ha npoAOBmeHHi CTOpoHH AC BiAKJia-
AeHo siApiaKH AD = AB i CE = CB (11aJI. 89). HK aHaATH
KYTH TPHKYTHHKa DBE, aHaIO'IH KYTH TPHKYTHHKa ABC?
40. Y TpHKYTHHKY OAHH a BHYTpi.mHix KYTiB AOPiBHIOe 30°, a OAHH
ia aOBHimHix 40°. 3aaiiAiTL pemTY BHYTPimHix KYTiB TPHKYT-
HHKa.
41. 3 sepmHHH npsMoro KyTa TpHKYTHHKa ABC nposeAeHo BHCOTY
BD. 3uaAAiTL KyT CBD, aHaIO'IH, ~o: 1) L A = 20°; 2) LA=
= 65°; 3) LA= a.
42. 3 sepmHHH Tynoro KYTa B TPHKYTHHKa ABC nposeAeHo BHCOTY
BD. 3mUiAiTL KYTH TPHKYTHHKiB ABD i CBD, aHaIO'IH, ~o
LA= a, LB= ft
43• .D;o:eeAin., ~o B llJ>SIMOICYTHOMY TpHKY'fHHKY 3 KYTOM 30°' KaTeT,
npoTHJiemHBii ~MY KYTY, AOpiBmoe noJIOBHHi rinoTeayau.
44. 3uaiiAiTL KYTH npgMOKYTHoro piBHo6eApeHoro TPHKYTHHKa.
45. Y piBHOCTOpoHHbOMY TPHKYTHHKY ABC npoBeAeHo MeAiaay
AD. 3aaiiAiTL KYTH TPHKYTHHKa ABD.
46. BucOTH TpHKYTHHKa ABC, npoBeAeHi a sepmHH A i C, nepe-
THHaIOTLCSI B TO~ M. 3uaiiAiTL L AMC, SIK~O L A = 70°,
LC= 80°.
47~ Y TpHKYTHBKY ABC MeAiaHa BD AOPiBHIOe noJioBHHi cropoHH
AC. 3aaii:AiTL KYT B TPHKYTHHKa.
48. 1Ipg11a a nepeTHHae BiApiaoK BC B iioro cepeAHHi. ,D;oseAiTh, ~o
TO'IKH B i C aHaxOABTbCSI Ha OAHaKosiii siAcTaHi siA nps11o'i a.
49. BiApiaoK BC neperHHae npSIMy a B TO~ 0. BiAcTS.Hi BiA TO'IOK
B i C AO npsMo'i a pisui• .D;oseAiTL, ~o TO'IKa 0 e cepeAHHOIO
siApiaKa BC.
50. ,D;oseAiTL, ~o BiACTaHi BiA 6yAh-SIKHX ABOX TO'IOK npRMOl AO
napaJieJILHOl IlpSIMOl piBHi.
51. ,D;ose.zi;iTL, ~o sip;CTaHi siA BepmHH piBHOCTopoHH1>oro TPHKYT-
HHKa AO npSIMHX, SIKi MiCTSITh npoTHJiemHi iM CTOpoHH, piBHi.
§ 5. I'EOMETPHqHI IIOBYAOBH
38. KOJIO
KoJtoM aaausaGTbCSI tl>irypa, S1Ka cKJiaAaGTbCSI a ycix TO'IOK
IlJIO~, piBHOBiAAaJieHBx Bi.A .zi;aaoi TO'IKH. D;SI TO'IKa HaaHBae-
ThCSI ijenrpoM KOJia.

7 KJiac 64
0 A
0
.-------...------..0/liaMemp
Ma.JI. 90 Man. 91
BiACT&Hb BiA roqoK KOJia AO ii:oro :u;eHTpa H83HB86Tbca pai}iycoM
KOJia; PaAiycoM Ha3HBa6Tbca T&Kom 6yAb-$1KHH BiApi3oK, i:u;o cno-
nyqae TO'IKY KOJia 3 ii:oro :u;eHTpoM (MaJI. 90).
BiApi30K, i:u;o cn?Jiyqae ABi TO'IKH KOJia, Ha3HBaeTi.ca xop0010.
XopAa, i:u;o npOXOAHTb '1epe3 :u;eHTp, H83HB86TbC$1 oiaMerpoM.
Ha MaJIIOHKY 91 BC - xopAa, AD' - AiaMeTp.
3 a A a 'I a (3). .rr;oBeAiTb, i:u;o AiaMeTP KOJia, aKHH
npoxoAHTb 'lepe3 cepeAHHY xopAH, nepneHAHKYJISpHHH
AO Hei'.
0
Man. 92
Po 3 u' a 3 a H Ha. Hexaii AB - xopAa
KOJI8 i c - i'i' cepeAHHa (M8JI. 92). TpH-
KYTHHK AOB piBHo6eAPeHHii 3 ocHoBolO
AB. y HbOMY CTOpOHH OA i OB piBHi
SK paAiYCH KOJia. 3a BJI8CTHBiCTIO MeAi-
aHH piBHo6eApeHoro TPHKYTHHKa, npo-
BeAeHoi' AO OCHOBH, BiApi3oK (JC 6
BHCOTOIO. ToMy AiaM~ KOJia, npoBeAeHHH
qepea cepeAHHY xopAH, nepneHAHKyJiap-
HHH AO XOPAH.
39. KOJIO, OIIHCAHE HABKOJIO TPHKYTHHKA
KoJIO H83HB86TbC$1 onucaHUM HaBKOJIO TpHKyTHHKa, $1Klll;O BOHO
npoxoAHTb qepea yci :Horo aepmHHH.
Teo p e Ma 5.1. .l(eHTP 'ICOJUI., on.ucaHoio Ha8'1Wll.O Tpu'ICyTHu-
'ICa, e T01'1'010 n.epeTUHY n.epn.eHiJu1Cyll.11pi8 iJo CTOpiH TpU1''flTHUKa,
npoBeiJeHux iiepea cef>eiJuHu ifux cropiH.