如图,在梯形ABCD中,AD‖BC,AB=CD,对角线AC、BD交于点O,点E、F分别在线段AO、DO上,且AE=DF，求证：四边形BE

显示全部楼层 · 2010-10-21 07:35:52

如图,在梯形ABCD中,AD‖BC,AB=CD,对角线AC、BD交于点O,点E、F分别在线段AO、DO上,且AE=DF，求证：四边形BEFG是等腰梯形

千问 · 2010-10-21 07:35:52

∵在梯形ABCD中,AD‖BC,AB=CD∴∠ABC=∠DCA,∠BAD=∠CDA∴ΔABC≌ΔDCB∴∠BAC=∠CDB∵AE=DF∴ΔABE≌ΔDCF∴BE=CF又∵∠BAC=∠CDB,而∠BAD=∠CDA∴∠OAD=∠ODA∴OA=OD,又AE=DF∴OE=OF∴∠OEF=∠OFE=1/2(180°-∠AOD)=∠OAD∴EF‖AD‖BC∴四边形BEFG是等腰梯形

千问 · 2010-10-21 07:35:52

∵在梯形ABCD中,AD‖BC,AB=CD∴∠ABC=∠DCA,∠BAD=∠CDA∴ΔABC≌ΔDCB∴∠BAC=∠CDB∵AE=DF∴ΔABE≌ΔDCF∴BE=CF又∵∠BAC=∠CDB,而∠BAD=∠CDA∴∠OAD=∠ODA∴OA=OD,又AE=DF∴OE=OF∴∠OEF=∠OFE=1/2(180°-∠AOD