• (2013?营口)如图,△ABC中,AB=AC,AD是△ABC外角的平分线,已知∠BAC=∠ACD.(1)求证:△ABC≌△CDA;(2)若∠B=60°,求证:四边形ABCD是菱形.试题及答案-解答题-云返教育

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      (2013?营口)如图,△ABC中,AB=AC,AD是△ABC外角的平分线,已知∠BAC=∠ACD.
      (1)求证:△ABC≌△CDA;
      (2)若∠B=60°,求证:四边形ABCD是菱形.

      试题解答


      见解析
      证明:(1)∵AB=AC,
      ∴∠B=∠ACB,
      ∵∠FAC=∠B+∠ACB=2∠ACB,
      ∵AD平分∠FAC,
      ∴∠FAC=2∠CAD,
      ∴∠CAD=∠ACB,
      ∵在△ABC和△CDA中
      {
      ∠BAC=∠DCA
      AC=AC
      ∠DAC=∠ACB

      ∴△ABC≌△CDA(ASA);

      (2)∵∠FAC=2∠ACB,∠FAC=2∠DAC,
      ∴∠DAC=∠ACB,
      ∴AD∥BC,
      ∵∠BAC=∠ACD,
      ∴AB∥CD,
      ∴四边形ABCD是平行四边形,
      ∵∠B=60°,AB=AC,
      ∴△ABC是等边三角形,
      ∴AB=BC,
      ∴平行四边形ABCD是菱形.
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