Геометрия7_9
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6.IIpose,n;iTh npRMy a. IIoaHaqre Ha Hi:a ,n;Bi .n;osiJibHi ToqKH A i
B. IIoaHa'!TeTenep J1Ky-He6y,n;L TO'IKYC Bi,n;pi3Ka AB.
7.ToqKa M nemHTb Ha npaMiii CD Mim TO'IKaMHCi 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 BHMipIOBaHHJI aa ,ll;OilOMOrOIO JiiHiHKH.
9.TpH TO'IKHA, 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'IKaMHBi C? qH Mome TO'IKaC nemaTH Mim TO'IKaMHAi B? SIRa a Tphox TO'IOKA, 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'IKHA, B, C nemaTb Ha o,n;Hiii npaMiH. qH Mome TO'!KaB poa- ,n;iJIJITH ToqKH Ai C, J1Km;o: AC= 7 CM, BC= 7,6 cM? Bi.n;nosi.n;b IlOJICHiTb.
12.qH MomyTh TO'IKHA, 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'IKHA, 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'IKHA, 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'aaRiBMae aa,n;aqa?
15.Ha Bi,n;piaKy AB ,n;oBmHHOIO 15 M B3JITO TO'IKYC. 3Haii.n;iTh ,n;oBmHHH 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;piaRiB 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 nepeTHHRB ii.
17.,Il;aHo npaMy i Tpu TO'IKHA, B, C, m;o He nemaTb Ha :u;i:H npaMiH. Bi.n;oMo, m;o Bi,n;piaoK AB nepeTHHae npaMy, a Bi.n;piaoK AC He
§ 1. OcuoaHi BJIBCTHBOCTi uaiinpocTiWHX reoMeTPH'IHHX <Piryp |
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nepeTHHa6 ii. qn nepeTnHa6 npsMy BmpiaoK BC? BiAnOBi,u;L
noscHiTL.
18.,IJ;aHo npaMy i 'IOTHPHTO'IKHA, 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) BmpiaKH 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'SITbTO'IOKi npaMa, m,o He npOXO,!J;HTb qepea >KOAHY 3 o;HX TO'IOK.BiAOMO, m,o TPH TO'IKHnemaTL B OAHiH niBnnom,uHi BiAHOCHO o;i6l npSIMOl, a ABi TO'IKH- B iHWiii niBilJIOID,HHi. KomHy napy TO'IOKCilOJiy'leHOBmpiaKOM. CKiJILKH BmpiaKiB nepe-rHHa6 npSIMy? BmnoBmL noacHiTL.
20.,IJ;aHo npsMy a i TO'IKHA, X, Y, Z Ha o;iii npRMi:H (Man. 11).
BmoMO, ID;O TO'IKHxi y JiemaTb no O,!J;HH 6iK Bm TO'IKHA, TO'IKHXi Z TaKom nemaTL no OAHH 6iK Bi,u; TO'IKHA. HK poaMi- 1.D;eHi TO'IKHY i Z Bi,!J;HOCHO TO'IKHA : no OAHH 6iK 'IHno piaHi 6oKn? IIoscHiTL BmnoBi,u;L.
21.IIoaHa'ITe,u;Bi TO'IKHA i B. IIpoBe,u;iTL niBnpsMy AB.
22.Ha Bi,u;piaKy AB B3SITO TO'IKYC. Cepe,u; niBnpSIMHX AB, AC, CA, CB HaaBiTL napy niBnpsMHX, aKi a6iraIOTLCSI, i napy ,u;onoBHHJibHHX niBnpaMnx. IIOSicHiTL BmnoBmL.
23.IIpoBe,u;iTL a o,u;His'iTO'IKHTpu AOBiJILHi npoMeHi. BnaHa'ITeHa OKO KYTH, YTBopeHi D;HMH npoMeHaMH. IIepeBipTe Barui Bi,u;noBi,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)? |
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26.Mim CTOpoHaMH KYTa (ab), m,o ,u;opiBHIOG 60°, npoxo,u;HTL npoMiHL e. 3Haii,u;iTL KYTH (ae) i (be), SIKID,O: 1) KYT (ae) Ha 30° 6iJILWHH aa KYT (be); 2) KYT (ae) B ABa pa3H 6iJibWHH 38 KYT
7 KJiac |
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(be); 3) npoMiHL c ,D;iJIHTL KYT (ab) IIOIIOJiaM; 4) |
rpap;ycHi MipH |
KYTiB (ae) i (be) Bi,ll;HOCSITLCSI, SIK 2 : 3. |
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27.IIpoBe,n;iTL 11pS1My. 1Io:ma11Te Ha ~ npS1MiH S1Ky-He6y,n;L TO'IKY A. IIoTiM Bi3LMiTL Ha OKO TO'IKYB Itie'i11pS1Mo'iTaK, ~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'IOKA, 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 (piBHoCTOpoHHiH TPHKYTHHK). IIepeBipTe To11HicTL 1106yp;OBH BHMip10BaHHS1M CTOpiH.
33.Ha cTopoHi AB TPHKYTHHKa ABC B3SITO TO'IKYD. l:JoMy .u;o-
piBHIOa CTOpoHa AB TpHKYTHHKa, SIK~o AD = 5 CM, a BD =
=6cM?
34.Ha cTopoHi AB TpHKYTHHKa ABC B3SITO TO'IKYD. 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'IHBi,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;opiBHI06 10 M, a KYT C = 90°. qoMy ,u;opiBHI06 cTopoHa PQ i KYT R? IIOSICHiTL Bi,u;IIOBi,u;L•.
39.TpHKYTHHKH ABC, PQR i XYZ piBHi. Bi,o;oMo, ~o AB = 5 cM,
§ 1. OcHOBHi BJiaCTHBOCTi HaiirrpocTiurnx reoMeTpH'lHHX <l>iryp |
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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.'IHMome 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. 'IHMOmHa rrpoBeCTH TpeTIO rrpSIMy, rrapaJieJibHY KOmHiH a ABOX AaHHX?
43.'IHMome rrpSIMa, w;o He rrpoxo,n;HTb 11epea moAHY a BepmHH TpHKYTHHKa, rrepeTHHaTH KOmHy :Horo CTOPOHy? qoMy?
44!' 1 ,D;aHo 'IOTHPHTO'IKH:A, B, Ci D. BiAoMo, w;o TO'IKHA, B, C JiemaTb Ha OAHiii rrpSIMiH i TO'IKHB, 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 'IOTHPHrrpSIMi a, b, c i d. Bi,u;oMo, w;o rrpSIMi a, b, c rrepeTHHaIOTbC.fl B o,u;HiH TO'IIVii rrpSIMi b, c, d TaKom rrepeTHHaIOTbCSI 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 rrepeTHHaIOTbCSI.
47!' ,D;aHo TpHKYTHHK ABC. Ha cTopoHi AC BaSITo TO'IKYB1, a Ha cTopoHi BC - TO'IKYA1. ,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 |
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Man. 28 |
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Man . 29 |
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Man. |
30 |
I 3ipO'IKOIO (*) 003Ha'leHO 38A8'1i niAsm:qeHOl CKJI8AHOCTi.
7 KJiac |
24 |
TbC.H B To11~i E. ,D.one,n;iT&, m;o ni,n;piaoK AC He nepeTHHae np.HMY BD (Ma.JI. 29).
49!' ,D.one,n;iTb, m;o KOJIH npoMiHb, JIKHH BHXO,ll;HTL 3 BepII!HHH KyTa, nepeTHHae ni,n;piaoK AB a KiH~MH Ha cropoHax KyTa, TO BiH nepeTHHae: 1) ni,n;piaoK AC a KiH~MH Ha cropoHax KyTa; 2) ,n;onin&HHH ni.zi;piaoK CD a KiH~BMH Ha cTopoHax KyTa (Man. 30).
50. ,D.one,n;iT&, m;o ,n;ni np.HMi a6o napanenLHi, a6o nepeTHHaIOT&cs B O,ll;Hiii TO'I~.
51!' To11KH A i C Ha.JiemaT& np.HMi:H a. Ha niBnps:Miii CA ni,n;Kna- ,n;eHo ni,n;piaoK CB, 6in&mHii aa niApiaoK CA. 1) HKa a Tp&ox TO'IOKA, B, C nemHT& Mim ,n;BoMa iHmHMH? IIoscHiT& niAnoBiA&. 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 'Ie 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;OilOBHaJibHHMH 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'IKaHa np.HMiH AB, m;o nemHT& Mi:>K TO'IKaMHA 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'laTKOBOIOTO'IKOIOc.
b
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8 |
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a, |
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Man. 31 |
Man. 32 |
§ 2. CyMiJKHi i sepTHK8JlbHi KYTH |
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Teo p e M a 2.1. |
CyMa cyMi.31cHux JCyTi6 iJopisff.K>e 180° . |
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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- |
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ropHyToro KyTa. ToMy CYMa KyTiB (a1b) i (a 2b) 11.opiBHIOO poaropHyTOMY 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 'Ia (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'IHMOrpaJJ.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:11eHU'4 3 nplUIUM lCYTOM, e npR.Mu4 JCyT.
KyT, MeHmHii aa 90°, HaaHeaeTbC.R zocrpuM JCyroM. KyT, 6iJILmHii aa 90° i MeHmHii eiJJ. 180°, HaausaeTbC.R rynuM.
L
rocmpuu KIJm
MaJI. 33
OcKiJibKH cyMa cyMiJKHHX KyTiB 11.opiBHIOe 180°' TO KyT, cyMiJKHHH a rocTPHM KyToM, TynHii, a KyT, cyMimmm a TynuM,- rocTpHii. Ha M&JIIOHKY 33 ao6pameHo TPH BHJJ.H KyTiB.
15. BEPTBKAJILHI KYTH
0 a Ha 'Ie.H Ha. Aea KYTH HaaHBaIOTbC.R BepruKaAbH.UMU,
.RKIIJ,O CTOPOHH OJJ.HOrO KyTa e lJ.OilOBH.RJibHHMH nienpaMHMH CTOpiH 11.pyroro.
7 KJiac |
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~h,
MaJI. 34 MaJI. 35
Ha MaJIIOHKY 34 KyTH (a1b1) i (a2b2) BepTHKaJibHi. CTopoHH a2 i b 2 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,l\CH
aaTeopeMOIO 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. |
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3 a A a 'Ia |
(9). CyMa ABOX KyTiB, YTBOpeHHx y peayJILTaTi |
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nepeTHHY ABOX npHMHX, AOPiBHIOe 50°. 3HaHAiTL ~i KYTH. |
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p 0 3 B'H 3 |
a H H s:. ,IJ;Ba KYTH, HKi YTBOpHJIHCH B peayJILTa- |
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Ti nepeTHHY )\BOX nps:MHX, a6o cyMi)l(Hi, a6o BePTHK8JILHi |
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(MaJI. 35). ,IJ;aHi KYTH He MO)l(YTb 6yTH cyMhKHHMH, OCKiJibKH |
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'ixcyMa ,l\OpiBHIOe 50°' a cyMa cyMi)l(HHX KYTiB CTaHOBHTb |
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180°. OT)l(e, BOHH BepTHKaJibHi. OcKiJii.KH BepTHKaJibHi KYTH |
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piBHi ii aa YMOBOIO lX cyMa 50°' TO KO)l(HHH 3 HHX AOPiBHI06 |
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25°. |
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16. IIEPIIEllNfKYJUIPHI IIP.JIMI |
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Hexaii a i |
b - |
ABi npHMi, ~o nepeTHHaIOTbCS: y TO'I~ A |
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(MaJI. |
36). KomHa a ~Hx npHMHx AiJIHTbcs TO'IKOIOA Ha ABi niB- |
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npaMi. IIiBnpaMi OAHisi'.npaMoi'. YTBo- |
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pIOIOTb 3 niBnpSIMHMH )\pyroi'. npHMOl |
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'IOTHPHKyTH. Hexa:H a - OAHH a ~x |
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KyTiB. BYAL-SIKHH 3 pellITH TpbOX KyTiB |
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6y)\e a6o cyMimHHM 3 KYTOM a, a6o |
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BepTHKaJlbHHM 3 KYTOM a. 3BiACH BH- |
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llJIHB86, ~o KOJIH OAHH 3 KyTiB npSIMHH, |
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TO pemTa KYTiB Te)I( npaMi. y ~bOMY |
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BHllaAKY roBOpSITb, ~o npaMi nepeTHHa- |
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MaJI. |
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IOTbCH niA npSIMHM KYTOM. |
§ 2. CyMhKHi i BepTHK&JILHi KYTH |
27 |
b |
c, |
A |
a |
a |
a, |
MaJI. 37 MaJI. 38
0 a Haq e H Hg. ,IJ;si npgMi Ha3HBaIOTbcg nepneniJU1cyJLapnu-
Mu, $1K~O BOHH nepeTHHaIOTbC$1 ni,u, np$1MHM KYTOM (MaJI. 37).
IlepneH,11,HKyJigpHiCT& np$1MHX Il03Ha'laIOTh3H8KOM _L. 3anHC a _L b 'IHT86TbC$1TaK: 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'IKaHa Hi:H. |
IloaHa'IHMOqepea 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 1b 1), ~o |
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,u,opiBHIOe 90°. To,u,i np$1Ma, gKa MiCTHTh npoMiHh bi, 6y,u,e nepneHAHKYJIHpHa ,u,o npaMoi a.
IlpHilYCTHMO, ~o icHye iHma npaMa, JIKa Te:>K npOX0,11,HTb qepea TO'IKYAi nepneH,U,HKyJiapua ,u,o npaMoi a. Iloaua11HMO 11epea c, nisnpaMy 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 a I·A.rre Bi,11, niBnp$1MOl a I y ,u,aHy
A
B a
MaJI. 39 |
MaJI. 40 |
7 KJiac |
28 |
niBilJIOil~llHY MOmHa BiAKJiaCTH TiJibKH OAHH KYT, mo AOPiBHI06 90°.
ToMy He Mome 6yTH iHmoi npsMoi, HKa npoxoAHTb 11epea TO'IKYA i nepneHAHKYJIHpHa AO npHMoi a. TeopeMy AOBeAeHo.
0 a H a 11 e H H H. llepnenauH:yJuipoM. AO AaHoi npaMoi HaaHBasTbCa Bi.u;piaoK npaMoi, nepneHAHKYJI&pHoi AO AaHoi npaMoi, BKHii Ma6 OAHHM ia CBOlX KiH~iB TO'IKYix nepeTHHY. IJ;eii KiHe~b BiApiaKa Ha3HBa6TbCJI OCH06010 nepneHAHKYJIHpa.
Ha MaJIIOHKY 39 nepneHAHKyJiap AB npoBe,u;eHo a TO'IKHA AO npaMoi a. To11Ka B - ocHoBa nepneHAHKyJiapa. ,lJ;JIH no6yAOBH nepneHAHKyJiapa KOpHCTYJOTbCJI KOCHH~eM (MaJI.
17. )J;OBE)J.EHHH BI)J. CYIIPOTHBHOrO
Cnoci6 AOBeAeHHa, aKHii MH aaCTocyBaJIH AO TeopeMH 2.3, Ha3HBa6TbCH AOBeAeHHJIM BiA cynpOTHBHOro. IJ;eii cnoci6 AOBeAeHH.H IlOJiaras B TOMy, mo Cil0'18TKYpo6HMO npHnymeHH.H, npOTHJiemHe TOMy, &Ke CTBepAmY6TbCH TOOpeMOIO. IIOTiM MiPKYB8HHHMH, CllHpaIO'IHCbHa aKciOMH i Bme AOBeAeHi TeopeMH, npHXOAHMO AO BHCHOBKY, mo cynepe'IHTba6o YMOBi TeOpeMH, a6o OAHiii 3 aKcioM, a6o AOBeAeHiii paHiIIIe TOOpeMi. Ha ~iii OCHOBi po6HMO BHCHOBOK, mo Harne npunymeuHa HenpaBHJibHe, a TOMY npaBHJibHe TBepAmeHHa TeopeMH.
IIoacHHMO ~e Ha npHKJiaAi AOBeAeHHa TOOpeMH 2.3. Y TeopeMi CTBepAmY6TbCH, mo qepea KOmHy TO'IKYnpHMOi 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- 'IHTb8KCiOMi Bi,u;KJiaAaHHH KyTiB, aa JIKOIO Bi,u; AaHoi niBnp.HMOi y AaHy niBDJIOmHHY MOmHa BiAKJI8CTH JIHWe OAHH KYT 3 AaHOIO rp8AYCHOIO Mipo10.
18. BICEKTPHCA KYTA
0 a H a q e H H a 1• BiceH:rpuco10 KYTa H83HB86TbC& npoMiHb, JIKHH BHXOAHTb 3 BepnrnHH KYT8, npoXOAHTb Mim iioro CTOpOH8MH i AiJIHTb KYT ll0IlOJI8M.
1 .D:aJii cnouo •oaHa'leHHH• aanHcyuaTH He 6y,u;eMo, a oaHa'lyBaHe noH.HTT.11 BHAiJIHTHMeMo KypcueoM.
§ 2. CyMilKHi i BepTRKSJihHi KYTR |
29 |
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Ha MaJIIOHKY 41 BH 6a'IHTeKYT (ab). |
a |
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IlpoMiHb c BHXO,ll;HTb 3 BepmHHH KyTa, |
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IIpoXO,ll;HTb Mim ii:oro cropoHaMH i ,zi;iJIHTb |
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KYT IIOilOJiaM: |
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L(ac)=L(bc)= L~ab). |
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3 a ,zi; a q a 17. ,ZJ;oee,zi;iTh, ~o 6i- |
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ceKTpHca KyTa YTBOpIOe 3 ii:oro CTOpo- |
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HaMH KYTH He 6i11bmi aa 90°. |
Man. 41 |
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Po a e'a a a H H a. HK MH aHae- |
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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'leHuareoMeTpii HeaHaHHa qoro-ue6y,zi;h a npoii:,zi;euoro MaTepia11y MOme CTaTH npH'IHHOIOuepoayMiHH$1 HOBOro MaTepi-
aJiy. HaBe,zi;eMo npHKJia,zi;.
IlpHIIYCTHMO, Ha ypo~i B'IHTeJibAOBO,ll;HTb TeopeMy npo piBHiCTb eepTHKaJihHHX KyTiB. HK DH aHaeTe, ,zi;na ,zi;oBe,zi;eHHa ~iei TeopeMH BHKOpHCTOBYIOTb 03Ha'leHH$1cyMimHHX KYTiB i TeopeMy rrpo CYMY cyMi)KHHX KyTiB. HK~o DH He aHaeTe, aKi KYTH Ha3HB8IOTbCa cyMi)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'leHoroMaTepiany. A ,zi;na ~oro cni,zi; noerop10BaTH
BHB'leHHHMaTepian aa KOHTPOJibHHMH 38IIHT8HHSMH.
IloBTOPIOBaTH rrpoii:,zi;eHHii MaTepiaJI 38 KOHTpoJibHHMH 38IIHTaHHRMH MomHa TaK. IIpo'IHTaii:TeaanHTRHHa. 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. |
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RopHCHo |
apo6HTH B~ pyKH MaJIIOHOK cl>irypH, oaaa11eHHa |
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a:Koi MaeTe ,zi;aTH. SIK~o y |
aanHTRHHi ii:,zi;eTbca rrpo TeopeMy, ccl>op- |
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MYJIIOii:Te ii, |
a'acyiiTe, B |
'IOMY nonarae YMOBa i BHCHOBOK Teo- |
peMH. |
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