Question

如图,在凸四边形abcd中,ac平分角bad,过c做ce垂直ab,垂足为e,且ae等于二分之一(ab+ad)

求证角abc+角adc等于180度

Answer 1

解:作CF垂直AD延长线于F.

∵∠AFC=∠AEC=90º;AC=AC;∠CAF=∠CAE.

∴⊿AFC≌⊿AEC(AAS),CF=CE;AF=AE.

∵AE=(1/2)(AB+AD).

∴AD+BE=AE=AF=AD+DF.

则BE=DF;又CF=CE.

∴Rt⊿CFD≌Rt⊿CEB(HL),∠CDF=∠B.

∴∠ABC+∠ADC=∠CDF+∠ADC=180º.