1. Geometry studies the properties of geometric figures. Basic figures include triangles, squares, and circles.
2. A point is a location in space. A line is infinite and straight. Three points that are not on the same line can be used to uniquely determine a plane.
3. A segment is the part of a line between two points. Its endpoints are called the vertices of the segment. Three points determine a unique plane.
КСУ опублікував Рішення стосовно пенсій «чорнобильців»Pravotv
КСУ визнав неконституційною норму закону, якою Верховна Рада уповноважила Кабінет Міністрів визначати мінімальні розміри пенсії за інвалідністю громадянам України, що постраждали внаслідок Чорнобильської катастрофи
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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,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
13. 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
14. § 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,
15. 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,
16. § 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
17. 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-
18. § 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
19. 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?
20. § 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.
21. 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
22. § 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
23. 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,
24. § 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.
25. 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
26. § 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.
27. 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.
28. § 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
29. 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.
30. § 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.
32. § 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
33. 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
34. 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.
35. 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-
36. § 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.
37. 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-
38. § 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).
39. 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.)
40. § 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.
41. 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
42. § 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•.
43. 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
44. § 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
45. 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.
46. § 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:.
47. 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.
48. § 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.
49. 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
50. § 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
51. 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
52. § 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
53. 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~
54. § 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
55. 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
56. § 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
57. 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.
58. § 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
59. 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
60. § 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.
61. 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-