The points P and Q have been taken on opposite sides AB and CD respect

1.	The points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD in such a way that AP = CQ (see figure).Show that AC and PQ bisect each other.
Answer» Given: ABCD is a parallelogram such that AP = CQ. To prove: AC and PQ bisect each other. ∠MAP = ∠MCP (alternate angles) AP = CQ (given) ∠APM = ∠CQM (alternate angles) ∴ ∆AMP ≅ ∆CMQ (by ASA congruency rule) ⇒ AM = CM and PM = QM (by c.p.c.t) ⇒ AC and PQ bisect each other.

The points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD in such a way that AP = CQ (see figure).Show that AC and PQ bisect each other.

Answer»

Given: ABCD is a parallelogram such that AP = CQ.

To prove: AC and PQ bisect each other.

∠MAP = ∠MCP

(alternate angles)

AP = CQ (given)

∠APM = ∠CQM

(alternate angles)

∴ ∆AMP ≅ ∆CMQ

(by ASA congruency rule)

⇒ AM = CM

and PM = QM (by c.p.c.t)

⇒ AC and PQ bisect each other.

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The points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD in such a way that AP = CQ (see figure).Show that AC and PQ bisect each other.

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