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Question
Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD such that AP = CQ (Figure). Show that AC and PQ bisect each other.
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Solution
Given: ABCD is a parallelogram and AP = CQ
To show: AC and PQ bisect each other.
Proof: In ΔAMP and ΔCMQ,
∠MAP = ∠MCQ ...[Alternate interior angles]
AP = CQ ...[Given]
And ∠APM = ∠CQM ...[Alternate interior angles]
∴ ΔAMP ≅ ΔCMQ ...[By ASA congruence rule]
⇒ AM = CM ...[By CPCT rule]
And PM = MQ ...[By CPCT rule]
Hence, AC and PQ bisect each other.
Hence proved.
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