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  • 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.

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 Prove: AC and PQ bisect each other.
Proof :

Here ABCD is a parallelogram
\Rightarrow AB\parallel DC
\Rightarrow AP\parallel QC
It is given that AP = CQ
Thus APCQ is a parallelogram.
And we know that diagonals of a parallelogram bisect each other.
Hence AC and PQ bisect each other.
Hence proved

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