Геометрия7_9
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31. 03HAKA IIAPAJIEJibHOCTI IIP.HMHX
Teo p e Ma 4.2 (oaHaKa napaJieJibHOCTi npBMHX). HICUfO
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MaJI. 73
§ 4. CyMa KyTi~ TPHKYTHHKa |
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MaJI. 75 |
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32. BJIACTHBICTb KYTJB,
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Teo p e Ma 4.3 (o6epHeHa .zi;o TeopeMH 4.2). Hx140 iJBi napalU!JibHi npR.Mi neperHyTi TpeTbOIO npR.M010, TO 6HyTpiWHi piaHOCTOpoHHi xyTU pi6Hi, a cyMa BHyTpiutHix oOHOCTOpoHHix xyTi6 iJopi6H10e 180°.
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§ 4. CyMa KyTiB TpHKyTHHKa |
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BiA ci11Hoi BC. |
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MaJI. 77 |
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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'IKHA i D JiemaJIH TIO piaHi 6oKH ei,n; TipHMOi BC (MaJI. 78).
KyTH DBC i ACB piBHi HK BHYTpimHi piaHOCTOpoHHi TIPH TiepeTHHi 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 TiapaJieJibHHX AC i BD i ci11Hoi AB, TO ix cyMa ,n;opiBHIOe 180°. TeopeMy
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A.n:e ~e HeMO>KJIHBO, 60 cyMa Bcix KYTiB TPHKYTHHKa p;opiBHI06 ID;o ii Tpe6a 6yJIO ,ll;OBeCTH.
3 a p; a 11 a 30. 1loMy p;opiBHIOIOTb KYTH piBHOCTOpOHHi.oro 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 H83HB86TbCH KYT' cyMi>KHHH a KYTOM TPHKYTHHKa npH aj.:H BepmHHi (MaJI. 79).
ID;o6 He nJiyTaTH KYT TpHKYTHHKa npH p;aHi:H BepmHHi ia aosHimHiM KYTOM npH aj.ii caMiH BepmHHi, :Horo iHop;i Ha3HBaIOTb 6HyrpiwniM KyroM.
Te o p e M a 4.5. 3osHimH.iu "YT |
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3 TeopeMH 4.5 BHilJIHBae, ~o aosHimH.iu "YT TpBlCYTHUJCa 6in.'b-
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3 a .a; a 11 a 35. Y TpHKYTHHKY ABC npoBep;eHo |
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CD. SlKa a Tpi.ox TO'IOKA, B, D Jie>KHTb Mim ,u;BoMa iHmHMH,
HK~o KYTH A i B TpHKYTHHKa rocTpi?
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MaJI. 79
§ 4. CyMa KyTiB TpHKYTHHK8 |
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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'IKaMHBi D. Orme, TO'lKaD nemHTL Mim TO'IKaMHA i B.
35. IlPSIMOKYTHHH TPHKYTHHK
TpHKYTHHK Ha3HBa6TLca npaMmcyrHuM, aK~o BiH Mas np.RMHii KyT.
OcKinLKH cyMa KyTill TpHKYTHHKa AOPiBHI06 180°, TO B npaMoKYTHOMY TPHKYTHHKY TinLKH OAHH np8MHH KyT. ,ll;Ba iHmHX KYTH npaMOKYTHoro TpHKYTHHKa rocTpi. CyMa rocTpHx KyTiB npaMoKyTHoro TPHKYTHHKa AOPiBHI06 180° - 90° = 90°.
CropoHa npaMOKYTHoro TPHKYTHHKa, ~o nemHTL npoTH npaMoro KyTa, Ha3HBaeTLC.R zinoreHy3010, p;Bi iHmi CTOpoHH Ha3HBaIOTbC.R
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Po a u'SI a a H HR. Hexaii ABC - |
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HHK 3 npSIMHM KYTOM c i KYTOM B, ~o ,o;opiBHIOG 30° (MaJI. 84). |
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Ilo6y,o;yeMo TPHKYTHHK DBC, ~o ,o;opiBHIOG TpHKYTHHKY ABC, |
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S1K noKaaaHo Ha MaJIIOHKY 84. Y TPHKYTHHKy ABD Bci KYTH |
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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 npHMo'ia, 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'IKHA AO npRMoi a.
,ZJ;oeeAeM.o 6.D;HHiCT& nepneHAHKYJI.Hpa AB. IlpBIIycTRMO icHyBaHH.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 |
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81 |
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Man. 85 |
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Man. 86 |
,IJ;oBJKHHa nepneap;HKyJisq>a, on~eaoro 3 p;aaoi TO'IKHHa 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'IOKnpaMOl AO napaJieJihHOl npHMOl plBHl. |
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Po 3 B'H 3 a a a H. Hexaii a i b - |
napaJieJihHi npHMi i A, |
A 1 - p;oBiJihHi TO'IKHHa npHMiii a (MaJI. 86). OnycTHMO 3 TO'l- KH A1 nepneap;HKYJIHI> A1B1 Ha npaMy b. Bi,u;KJiap;eMo Dip; TO'IKHB1 Ha npodii b Bi.n;piaoK B1B, m;o p;opiBHIOO Bi,u;piaKy
AA 1 TaK, m;o6 TO'IKHA 1 i B 6yJIH no piaai OOKH Bi.n; npHMoi
AB1. Top;i TpHKYTHHKH AB1A1 i B1AB piBHi 3a nepmo10 03HaK010. Y HHX CTOpoHa AB1 cniJILHa, AA1 = BB1 3a no6yp;o- BOIO, a KYTH B1AA1 i AB1B piBHi HK BHyTpimHi piaHocropoaHi 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'IHHXcl>iryp JIIO,D;H
.n;iCTaJIH B peayJILTaTi npaKTH'IHOlp;iaJihHOCTi i CDOCTepeJKeHb Hap; HaBKOJIHWHiM CBiTOM. 3 11acoM y11eai noMiTHJIH, m;o p;eHKi BJiaCTHBOCTi reoMeTpH'IHHXcl>iryp MOJKHa BHBecTH 3 iHWHX BJiaCTHBOCTeii MipKyBaHHHMH. TaK BHHBKJIH TeopeMH i p;osep;eHHH.
3'HBHJIOCJInpHpop;He 6amaliHH no MOJKJIHBOCTi 3MeHWHTH KiJILKiCTL THX BJiaCTHBOCTeii reoMeTPH'IHHX cl>iryp, HKi BHDJIHBalOTh 6e3nocepep;HLO 3 ,u;ocBip;y. TBepp;meHHa, ·m;o 3aJIHWHJIHCH 6e3
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58 |
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BHTKY ,ZJ;OCarJia oco6JIHBO BHCOKOrO piBHR |
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e SranTi. Y nepmoMy THCSAOJii'M'iAO Hamoi |
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epH reOMeTpH'IHi ei,ZJ;OMOCTi Bi,ZJ; erHilTRH |
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nepeihuJIH AO rpeKie. 3a nepioA a VII no |
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MeTpH He JIHme a6araTHJIH reoMeTpi10 |
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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 |
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yaaraJibHeHa EBKJiiAOM (330-275 pp. AO |
H. e.) e iioro aHaMeHHTiii npa~ cHaqaJia•.
BHKJiaA reoMeTpii e cHaqaJiax• EBKJii,ZJ;a no6y,ZJ;oeaHo Ha cHCTeMi aKcioM. IJ;a CHCTeMa ei,ZJ;pi3HR6TbCR ei,ZJ; CHCTeMH aKcioM, npHHHRTOl 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 MaTeMaTHKH 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 TBepAmeHHa, 6ye eeJIHKHii pociiicbKHii M&TeM&THK MHKoJia leaHOeHq JI06aqeec1>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 OAHOCTOpoHHiMH. 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'HiiirrpHMiii, MOmHa npoBeCTH napaJieJibHY iH npHMy. CKiJibKH npHMHX,· napaJieJibHHX p;aHiii, MomHa npoBecTH 11epea To11Ky, ~o He JiemHTb Ha ~iii npsMiii?
7.,ll;oBep;iTb, ~o KOJIH ABi napaJieJibHi npHMi nepeTHHaIOTbCS TpeTbOIO 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, napaJieJII>Hi. HK~o npHMa nepneH,IJ;HKyJispHa AO op;Hiei a p;Box napaJieJibHHX 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 TPHKYTHHKa?
16.HKa ia CTopiH npHMOKYTHOro TpHKYTHHKa HaaHBaeThcs: rin0Teuyao10? 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 TiJibKH OAHH.
19.~o Ha3HBa6TbCS: Bip;cTaHHIO TO'IKHBi,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 napaJieJibHHX 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.