Геометрия7_9
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MaJI. 13
PoarmrneMo Tenep nisnpHMi a no11aTKOBOIO TO'IKOIOC (nisnpHMi CA i CB). To11Ka C poa,zdn:He TO'IKHA i B. ToMy TO'IKHA i B He MO)KYTL HaJiemaTH O,ll;HIB nisnpHMIB, OTme, nisnp.11Mi
CA i CB ,ll;OilOBH$1JILHi.
7. KYT
KyroM HaausaeTLCH cl>irypa, HKa cKJia,n;aeTbCSI a TO'IKH- 6epwunu Kyra i ,ll;BOX pi3HHX nisnpSIMHX, w;o BHXO,ll;SITL i3 ~iei TO'IKH,- cropin Kyra.
Ha MaJIIOHKY 14 BH 6a11uTe KYT a sepmHHOIO 0 i cTopoHaMH a, b. KyT noaHa11aIOTL a6o iioro sepmHHOIO, a6o iioro cTopoHaMH, aoo aanHCYIOTL TpH TO'IKH:BepmHHY i ,ll;Bi TO'IKH,Ill;O Jie:>KaTL Ha cropoHax KyTa. CJioBo •KYT• iHo,n;i aaMiHSIIOTL aHaKoM L. KYT Ha MaJIIOHKY 14 MomHa nooHa'IHTHTpLoMa cnoco6aMu:
L AOB. Y TpeTLOMY cnoco6i noaHa11eHHS1 KyTa 6yKsa, w;o noaHa'lae sepm'HHYKyTa, CTaBHTLC.11 BCepe,n;HHi.
HKw;o cTopoHH KyTa e ,n;onoBHSIJILHHMH nisnpSIMHMH o,n;Hiei np.11Moi, TO KYT HaausaIOTL poazopnyruM. Ha MaJIIOHKY 15 au 6a'IHTe poaropHyTuii KYT a sepmHHOIO 0 i cTopoHaMH OA i OB.
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MaJI. 14 |
MaJI. 15 |
MaJI. 16 |
§ 1. Ocaosai BJiacTHBOCTi aaii:npocTiWHx reoMeTPH'IHHX«l>iryp |
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By,11;eMo roBopHTH, m;o npoMiHb npoxoiJurb Mi:HC cTopoHaMH |
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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 'Ia 25. qH MO)Ke npoMiHb c npOXO,ll;HTH Mim CTOpo- |
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HaMH KyTa (ab), J1Km;o L (ac) = 30°, L (cb) = |
80°, L (ab)= |
= 50°? |
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Po a e'JI a au H JI. HKm;o npoMiHb c npoxo,zi;HTb Mim cTopouaMH KyTa (ab), TO aa BJI8CTHBiCTIO BHMipIOBaHHJI KyTiB Mae 6yTH L (ac) + L (cb) = L (ab). AJie 30° + 80° =F 50°.
OTme, npoMiHb c He Mome npoxo,zi;HTH Mim cTopoHaMH KyTa (ab).
MaJI. 17
7 KJiac |
12 |
8. BI,D;KJIA,D;AHH.H BI,D;PI3KIB I KYTIB
Ha MaJIIOHKY 18 noKa3aHo, HK 3a ,n;on0Moro10 JiiHiHKH Ha niBnpHMiH a 3 noqaTKOBOIO TO'lKOIO A MOJKHa Bi,ll;KJiaCTH Bi,n;pi3oK ,n;aHOl ,ll;OBJKHHH (3 CM).
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MaJI. 18
Ilo,n;HBiTbCH Ha MaJIIOHOK 19. IliBnpHMa a, npO,ll;OBJKeHa 3a noqaTKOBY 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°). |
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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: |
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VI. Ha 6yiJb-1uciu niBnpaMill Bia ii no11aT1to8o'i To•ucu Mo:NCna |
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BiiJJCJr.acTU BiiJpi301C iJaHO'i iJ08:11CUHU i TillblCU OiJUH.
VII. Bia 6yiJb-1'1CO'i niBnp1u,o'i y aany nienllOUfUHY MO:llCKa eiiJJCJr.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'lOKA, Bi C JieJKHTb
§ 1. OcHOBHi BJiaCTHBOCTi Hai1:npocTiwHx reoMeTpH'IHHX<liiryp |
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p 0 3 B'H 3 a H H H |
(MaJI, |
20). |
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OcKiJibKH TO'lKHB i C Jie:>KaTb Ha oAHiii |
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ni.JmpHMiii a no'laTK0~010 TO'IKOIOA, |
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TO BOHH He poa,a;iJIHIOTbCH TO'IKOIOA, |
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TOOTO TO'IKaA He Jie:>KHTb Mi:>K TO'IKa- |
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MH Bi c. |
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qH MO:>Ke |
TO'IKa B Jie:>KaTH |
Mi:>K |
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TO'IKaMHA i |
C? HK6H BOHa Jie:>KaJia |
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Mi:>K TO'IKaMHA i C, TO 6yJIO 6 AB + |
Man. 20 |
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+BC= AC. AJie ~e HeMO:>KJIHBo, |
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ocKiJibKH aa YMOBOIO Bi.ApiaoK AC MeHmHii aa Bi.a;piaoK AB. |
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0Tme, TO'IKaB He Jie:>KHTb Mim TO'IKaMHA |
i |
C. |
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3 Tpbox TO'IOKA, B, C OJJ;Ha Jie:>KHTb Mi:>K ABOMa iHIIIHMH. |
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ToMy TO'IKaC Jie:>KHTb Mim TO'IKaMHA i B. |
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9. TPHKYTHHK |
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TpuKyrnuJCoM HaaHBaGTbCH <Pirypa, HKa cKJia)J;aGTbCH a Tpbox |
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TO'IOK, ~o He Jie:>KaTb Ha |
O,ll;HiH npHMiH, i TpbOX |
Bi.ApiaKiB, HKi |
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nonapHo cnony'laIOTbItj. TO'IKH. To'IKHHa3HBaIOTbCH 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 eepIIIHHaMH. 3aMiCTb cnoBa •TPHKYTHHK• iHOAi B:>KHBaIOTb aHaK t:,.. HanpHKna.a;, TPHKYTHHK Ha MaJIIOHKY 21 noaHa'laeTbCHTaK:
KyroM rpuKyrnuKa ABC npH eepmHHi A H83HBaGTbCH KyT, YTBOpeHHii nienpHMHMH AB i AC. TaK caMo oaHa11a10TbCH KYTH
TPHKYTHHKa npH eepmHHax Bi C.
,ll;Ba Bi.Api3KH H83HB8IOTbCH pi6HUMU, HK~O BOHH MaIOTb O)J;H8KO-
B
[
A |
[ |
A |
B A, |
B, |
Man. 21 |
Man. 22 |
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 3BHqaiiHHM 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- |
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SICHiTL. |
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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). IlepeMiCTHMO 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 npoAOB)l(0HH$1. BepmHHH Hamoro TPHKYTHHKa y ~MY HOBOMy llOJIO- )l(eHHi noaHaqHMO Ai. Bi. C1 (MaJI. 23, 6).
TpUICYTHUK AiBiC1 aopiBHJOe rpu-,cyrnu-,cy ABC.
lcHyBaHHH TPHKYTHHKa A1BiCi, ~o p;opiBHIOG TPHKYTHHKY ABC i poaMi~eHHii 3a3HaqeHHM cnoco6oM BiAHOCHO p;aHoro npoMeHH a,
§ 1. OcHOBHi BJiaCTHBOCTi HaiinpocTiUIHX reo:r.i:eTpH'IHHXcl>iryp |
15 |
a
[
A |
B |
a) |
0) |
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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 nepeTHHRIOTbCB.
Ha MRJIIOHKY 24 noKaaaHo, BK aa AOnoMoroio KYTHHKa i JiiHiiiKH npouecTH 11epea ARHY TO'IKYB npBMY b, napa11e111>Hy ARHiii npsMiii a.
Ilapa11e111>HicT1> npsMHX noaHa11a10T1> aHaKoM II. 3anHc a II b 'IHTaeT1>cs: •IlpsMa a napa11e111>Ha nps:Miii b•.
A
a
b
Man. 24 |
Man. 25 |
7 KJiac |
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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. |
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r1\ |
3 a A a q a (41). qH MO:llCe npS114a, ID;O nepeTHHae OAHY 3 |
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.u;~ox n~paneJibHHX np.RMHX, He nepeTHHaTH .u;pyry? IIo.acHiTb |
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Bl.ll;llOBl.n;b. |
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Po a B'.a a a H H s. |
Hexaif a i b - |
napaJieJibHi np.aMi i |
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Hexaif np.RMa c nepeTHHae np.aMy a |
B To11u;i A (MaJI. 25). |
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KoJIH 6 np.aMa c He nepeTHHaJia np.aMy b, TO 11epea TO'IKYA |
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npoxo,n;HJIH 6 ABi np&Mi, ID;O He nepeTHHRIOTb np.RMY b: |
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np.aMa a i np.aMa c. AJie aa BJiacTHBiCTIO napaJieJibHHX np.RMHX |
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u;e HeMOlKJIHBO. 0Tate, np.aMa c, .RKa nepeTHHae np.RMY a, |
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IlOBHHHa nepeTHHRTH i |
np.RMY b, napaJieJibHY lli:. |
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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 MipKyBaHH.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. 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'IKaC JiemHTb B o,n;Hiif nienJIOID;HHi a TO'IKOIOA, TO ui.u;pi3oK AC He nepeTHHae npaMy a, a Bi.a;pi3oK BC nepeTHHae u;ro npsMy (Man. 26, a).
HKID;O TO'IKaC nemHTL B o.a;Hi:H niunnoID;HHi a TO'IKOIOB, 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'IRHHOCKJia,n;aeTLC.R 3 ABOX 11acTHH.
B o,n;Hi:H 11aCTHHi rouopHTLCB npo Te, ID;O .a;aHo. :u;s 11acTHHa Haau-
§ 1. 0cHOBHi BJiaCTHBOCTi Haii.npocTiWmt reoM0TpH'IHHXcl>iryp |
17 |
B
a) |
0) |
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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'a6B TOMy, ~o u;a npaMa nepeTHHa6 TiJlbKH O,ll;HY 3 ,ll;BOX illmHX CTopiH TPHKYTHHKa.
13. AKCIOMH
Tsep,11;meHHR, RKi MiCTRTb cl>opMyJIIOBaHHR OCHOBHHX BJiaCTHBOCTeii HaiinpocTimHX «l>iryp, He ,ll;OBO,ll;RTbCSI i Ha3HBalOTbCH a1'CiOMaMU. 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 aKCiOMaMH, 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'IHCbHa aKCiOMH i reopeMH, ,ll;OBe,11;eui paHime.
y reoMeTpi'inopa,11; 3 TaKHMH CJIOB8MH, RK aKcioMa i TeopeMa, BlKHBaeTbCSI TaKOm CJIOBO •03H8'1eHHSI•. ,lJ;aTH 03Ha'teHHSI. qoroHe6yp;h 03Ha11ae IIORCHHTH, ~o u;e TaKe.
'HanpHKJia,11;, rosopn&: •AaiiTe oaHa'leHHaTPHKYTHHKa•. Bi,11;- noBi,D;aIOTh: •TpHKyTHHKOM Ha3HB86ThCSI cl>irypa, RKa CKJI8,11;86TbCSI
7 KJiac |
18 |
3 TpbOX TO'IOK,~o He Jie:>KaTb Ha op;Hiii npaMiii, i TpbOX Bip;piaKiB, SIKi nonapHO CIIOJIY'IRIOTb~ TO'IKH•.
,l.J;pyrHii npHKJiap;: •,l.J;aiiTe 03Ha'leHHSInapaJieJibHHX npSIMHX•. Bi.zi;nosiAaIOTL: dlpaMi HaaHBaIOTbCSI napaJieJibHHMH, .HK~o BOHH He nepeTHHaIOTLca•. BH Bme aHasTe oaHa11eHHS1 piBHOCTi Bi.zi;piaKiB, piBHOCTi KYTiB i TPHKYTHHKiB.
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KOHTPO~hHI3AilHTAHH.H |
• |
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1.HaBep;iTL npHKJiap;H reoMeTpH'IHHXcpiryp.
2.HaaBiTL ocHOBHi reoMeTpH'IHicpirypH Ha IIJIO~HHi.
3.HK rroaHa11aIOTLCS1 TO'IKHi 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 BiAHOCHO 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 03H8'186BHpaa: tlliBrrpaMa rrpoxo,n;HTb Mi:>K CTOpOHSMH KyTa •.
17.B SIKHX OAHHHIJ,SIX BHMipIOIOTb KYTH i aa AOIIOMOrOIO SIKOro iHCTpyMeHTa? IIoacHiTb, .HK BHKOHYIOTb BHMipIOBSHH.H.
18.CcpopMyJIIOiiTe ocHOBHi BJiaCTHBOCTi BHMip10BaHHa KyTiB.
19.CcpopMyJIIOHTe ocHoBHi BJISCTHBOCTi ui,n;KJiap;aHH.H ui.zi;piaKiB i KYTIB.
20.IIJ,o TaKe TPHKYTHHK?
21.IIJ,o TaKe KYT TpHKYTHHKa rrpH ,n;aHiH BepurnHi?
22.HKi Bi,n;piaKH Ha3HB8IOTbCSI piBHHMH?
§ 1. OcHOBHi BJiaCTHBOCTi HaHnpocTiurnx reoMeTPH'IHHX «Iiiryp |
19 |
23.HKi KYTH HaaHBaIOTLC.11 piBHHMH?
24.HKi TPHKYTHHKH HaaHBaIOTLc.11 piBHHMH?
25.HK noaHa11aIOTL Ha MaJIIOHKY Bi,11;n0Bi,11;Hi cTopoHH i KYTH piBHHX TpHKYTHHKiB?
26.Ilo.11cHiTL aa MaJIIOHKOM 23 icHyBaHH.11 TPHKYTHHKa, ru;o ,11;opiBHI06 ,D;aHOMy.
27.HKi np.11Mi HaaHBaIOTLC.11 rrapaJieJILHHMH? HKHM aHaKoM rroaHa11aroTL rrapaJieJILHicTL rrp.11MHx?
28.Cq>0pMyJiroiiTe ocHOBHY BJiacTHBiCTL rrapaJieJILHHX rrp.llMHX.
29.HaBe,11;iTL rrpHKJia,11; TeopeMH•
3AAAq1 1
1. 1) IlpoBe,11;iTL np.11My. IloaHa11Te .11Ky-He6y,11;L T0111$Y A, ru;o |
Jie- |
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•lKHTL Ha rrp.11Miii, i |
TO'IKYB, ru;o He JielKHTL Ha rrp.11Miii. |
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2) IlpoBe,11;iTL p;Bi rrp.11Mi a i |
b, ru;o rrepeTHHaIOTLC.11. IIoaHa11Te |
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TO'IKY C rrepeTHHY |
rrp.11MHx; |
TO'IKYA Ha rrp.11Miii a, |
ru;o |
He |
JielKHTL Ha rrp.11Miii |
b; TO'IKYD, ru;o He JielKHTL Ha |
mop;Hiii |
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3 rrp.llMHX a i b. |
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2.IloaHa11Te Ha apKywi rrarrepy ,D;Bi TO'IKH.IlpoBep;iTL 11epea HHX rrp.11My Bi.A pyKH. 3a p;on0Moro10 JiiHiiiKH rrepeBipTe rrpaBHJILHiCTL rro6yp;OBH.
3.llH MOlKYTL p;Bi rrp.11Mi MaTH p;Bi TO'IKHrrepeTHHy? 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'IKHaHoBy rrpoBOA.llTL JiiHiro. HKru;o JiiHil He a6irJIHc.11, TO JiiHiiiKa 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'IKHA B. __','_1Li
IloaHa11Te Terrep TO'IKYC TaK, ru;o6
TO'IKaA JiemaJia Mim TO'IKaMHBi C.
1 B8r8TO 3~311 ~LOro n~py11HHK8 B3.RTO i3 WKiJibHHX
i 3~8'1HHKiB MHHYJIHX poKiB, 30KpeM8 3 .reoMeTpih A. n. KuceJibOB8 i •36ipHHK8 3~8'1 3 reoMeTpih M. 0. Pn6KiH8.