geometry

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in triangle ABC, PandQ are points on BC such that BQ=CP and AP=AQ.show thatAB=AC

  • geometry -

    In this case, APQ forms an isoceles triangle, which you'll see if you draw a figure. (AP=AQ)

    So, ∠APQ = ∠AQP
    => 180 - ∠APQ = 180 - ∠AQP
    => ∠AQB = ∠APC .....1

    Now, consider triangles AQB and APC
    AP = AQ (Given in the question)
    ∠AQB = ∠APC (From ...1)
    QB = PC (Given in the question)

    Hence, the two triangles AQB and APC are congruent by the SAS rule.

    Since the corresponding parts of congruent triangles are equal, AB=AC

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