Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD such that AP=CQ. Show that AC and PQ bisect each other.

Explanation

Given ABCD is a parallelogram and AP = CQ
To show AC and PQ bisect each each other.

Proof In `Delta`AMP and `Delta`CMQ,
`angleMAP=angleMCQ" "`[alternate interior angles]
AP=CQ`" "`[given]
and `" "angleAPM=angleCQM" "`[alternate interior angles]
`therefore" "DeltaAMP~=DeltaCMQ" "` [by ASA congruence rule]
`rArr" "AM=CM" "`[by CPCT rule]
and `" "PM=MQ" "`[by CPCT rule]
Hence, AC and PQ bisect each other. `" "` Hence proved.