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1.3 探索三角形全等的条件(1)-(8)答案-苏科版八年级上册数学补充习题答案

1.3 探索三角形全等的条件(1)

1、△ACB ≌ NMR,△DEF ≌ △QOP.
2、在△ABC和△CDA中,
∵AB = CD, ∠BAC= ∠DCA,
AC = CA,
∴△ABC ≌ △CDA(SAS).
3、∵AB ⊥ CD,∠ABC = ∠DBE = 90°.又
AB = DB,BC = BE,
∴△ABC ≌△DBE(SAS).
4、(1) ∵AD = AE, ∠1 = ∠2, AO = AO,
∴△AOD ≌ △AOE( SAS).
(2) ∵AC = AB,∠1 = ∠2, AO = AO,
∴△AOC ≌ △AOB( SAS).
(3) ∵AB = AC,∠BAD = ∠CAE,AD = AE,∴△ABD ≌△ACE( SAS).

1.3 探索三角形全等的条件(2)

1、∵ AD是△ABC的中线,
∴ BD = CD.又∠BDN = ∠CDM,
DN = DM,
∴ △BDN ≌ △CDM( SAS).
2、∵ AD是△ABC的中线,
∴BD = CD.
∵ AD ⊥ BC,
∴∠ADB = ∠ADC = 90°.在△ABD和
△ACD中,
∵AD = AD,∠ADB = ∠ADC, BD = CD,
∴△ABD ≌ △ACD(SAS).
∴ AB = AC.
3、在△ABC和△DEF中,
∵AB = DE, ∠B = ∠E, BC = EF,
∴△ABC ≌ △DEF(SAS).
∴ ∠ACB = ∠DFE.
∵∠ACF + ∠ACB = ∠DFC + ∠DFE = 180°,
∴ ∠ACF = ∠DFC.
∴ AC ∥ DF.
4、(1) 利用(SAS)证明;
(2) 共可画14条.

1.3 探索三角形全等的条件(3)

1、∵ AB ∥ DC,AD ∥ BC,
∴ ∠BAC = ∠DCA,∠BCA = ∠DAC.
在△ABC和△CDA中,
∵∠BAC = ∠DCA,AC = CA,
∠BCA = ∠DAC,
∴ △ABC ≌ △CDA(ASA). ∴ AB = DC,
AD = BC.
2、在△ABE和△ACD中,
∵∠A = ∠A,AB = AC,∠B = ∠C,
∴ △ABE ≌ △ACD(ASA).
∴ AD = AE.
∴ AB - AD = AC - AE.即DB = EC.
3、∵ ∠3 + ∠AOB = ∠4 + ∠AOC = 180°,∠3 = ∠4,
∴∠AOB = ∠AOC.在△AOB和△AOC中,
∵ ∠1 = ∠2, AO = AO,∠AOB = ∠AOC,
∴ △AOB ≌ △AOC(ASA).
∴ OB = OC.

1.3 探索三角形全等的条件(4)

1、∵ AB ∥ CD,
∴ ∠ABE = ∠CDF.
∵ AE ⊥ BD,CF ⊥ BD,
∴ ∠AEB = ∠CFD = 90°.
在△ABE和△CDF中,
∵ ∠ABE = ∠CDF,∠AEB = ∠CFD,
AE = CF,
∴ △ABE ≌ △CDF(AAS).∴ AB = CD.
2、∵ △ABC ≌ △DCB,
∴ AB = DC,∠A = ∠D.在△AOB和△DOC中,
∵ ∠A = ∠D,∠AOB = ∠DOC,AB = DC,
∴ △AOB ≌ △DOC(AAS).
3、(1) 在△ABE和△ACD中,
∵ ∠A = ∠A,∠B = ∠C,AE = AD,
∴△ABE ≌ △ACD(AAS).
(2)∵△ABE ≌ △ACD,
∴ AB = AC,AB - AD = AC - AE,即DB = EC.在△BOD和△COE中,
∵ ∠DOB = ∠EOC,∠B = ∠C, DB = EC,
∴ △BOD ≌ △COE(AAS).

1.3 探索三角形全等的条件(5)

1、∵ B是EC的中点,
∴ BE = BC.
∵ ∠ABE = ∠DBC,
∴∠ABE + ∠ABD = ∠DBC + ∠ABD,
即∠DBE = ∠ABC.在△DEB和△ACB中,
∵ ∠DBE = ∠ABC,∠D = ∠A,
BE = BC,
∴ △DEB ≌ △ACB( AAS).
∴DE = AC.
2、∵ CD ⊥ AB,EF ⊥ AB,
∴ ∠CDB = ∠EFA = 90°,
∵ AD = BF,
∴ AD + DF = BF + DF,即AF = BD.在△CBD和△EAF中,
∵ CD = EF, ∠CDB = ∠EFA,BD = AF,
∴△CBD ≌ △EAF(SAS).
∴∠A = ∠B.
3、∵ ∠AFB = ∠AEC,∠B = ∠C,AB = AC,
∴ △ABF ≌ △ACE(AAS).
∴ ∠BAF = ∠CAE.
∴ ∠BAF - ∠EAF = ∠CAE - ∠EAF,即∠BAE = ∠CAF.

1.3 探索三角形全等的条件(6)

1、连接BD.
∵ AB = CB, AD = CD,
BD = BD,
∴ △ABD ≌ △CBD(SSS).
∴ ∠A = ∠C.
2、∵AB = DC,AC = DB,BC = CB,
∴ △ABC ≌ △DCB(SSS).
∴ ∠ABC = ∠ DCB,∠ACB = ∠DBC.
∴ ∠ABC - ∠DBC = ∠DCB - ∠ACB,
即∠1 = ∠2.
3、△ABC ≌ △CDA( SSS),△ABE ≌ △CDF( SAS),
△ADF ≌△CBE(SAS).证明略.

1.3 探索三角形全等的条件(7)

1、(1) 图略;
(2) 在△OPE和△OPF中,
∵ ∠EOP = ∠FOP,OP = OP,
∠OPE = ∠OPF= 90°,
△OPE ≌△OPF(ASA).
∴ PE = PF.
2、(1) 图略;
(2) 在△OPM和△OPN中,
∵ ∠MOP = ∠NOP,∠PMO =
∠PNO = 90°,OP = OP,
∴ △OPM ≌ △OPN(AAS).
∴ PM = PN.

1.3 探索三角形全等的条件(8)

1、∵ AB ⊥ BD, CD ⊥ DB,
∴ ∠ABD = ∠CDB = 90°,在Rt△ABD和
Rt△CDB中,
∵ AD = CB, DB = BD,
∴ Rt△ABD ≌ Rt△CDB( HL).
∴ AB = CD.
2、在Rt△ABF和Rt△DCE中,∠B = ∠C

= 90°,AF = DE,AB = DC,
∴ Rt△ABF ≌ Rt△DCE( HL).
∴ BF = CE.
∴ BF - EF = CE - EF,即BE = CF.

3、在Rt△ADE和Rt△ADF中,
∵ ∠AED = ∠AFD = 90°,DE = DF,AD = AD,
∴ Rt△ADE ≌ Rt△ADF( HL).
∴ ∠EAD = ∠FAD.在△ADB和△ADC中,∠ADB = ∠ADC = 90°,AD = AD,
∠BAD = ∠CAD,
∴ △ADB ≌△ADC(ASA).
∴ AB = AC.
4、在Rt△ADB和Rt△BCA中,
∵ ∠ADB = ∠BCA = 90°.BD = AC, AB = BA,
∴ Rt△ADB ≌ Rt△BCA(HL).
∴ AD = BC.在△ADC和BCD中,
∵ AC = BD,AD = BC,DC = CD.
∴△ADC ≌ △BCD.
∴ ∠2 = ∠1.

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