Question

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.

Answer 1

Given:

$ABCD$ is a parallelogram and $AP=CQ$

To show : $AC$ and $PQ$ bisect each other.

Proof:

in $\Delta AMP$ and $\Delta MCQ$ [alternate interior angles]

AP = CQ [ given]

And $\angle APM=\angle CQM$ [alternate interior angles]

$\therefore \Delta AMP\cong \Delta CMQ$ [ By ASA congruence rule]

$\Rightarrow AM=CM$ [ By CPCT rule]

And $PM=MQ$ [ By CPTC rule]

Hence, $AC$ and $PQ$ bisect each other.