1.

Point P and Q are taken on the opposite side AB and CD respectively of a parallelogram ABCD such that AP = CQ. show that AC and PQ bisect each other.

Answer»

Let point of intersection of AC and PQ be E.
Taking triangle AEP and CEQ.
AP = CQ (GIVEN)
ANGLE EAP = angle ECQ (alternate interior angles are ALWAYS equal)
angle EPA = angle EQC (alt. int. angles).
So, by ASA triangle AEP is congruent to triangle CEQ.
By c.p.c.t.c, AE =EC and PE = EQ


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