In parallelogram $ABCD$,two points $P$ and $Q$ are taken on diagonal $BD$ such that $DP = BQ$ (see figure). Show that $APCQ$ is a parallelogram.

(N/A) We have parallelogram $ABCD$. $BD$ is a diagonal and points $P$ and $Q$ are such that $DP = BQ$ [Given].
To prove that $APCQ$ is a parallelogram.
Let us join $AC$ intersecting $BD$ at $O$.
Since the diagonals of a parallelogram bisect each other,we have $AO = CO$ and $BO = DO$ ... $(1)$.
We are given $DP = BQ$ ... $(2)$.
Subtracting equation $(2)$ from $BO = DO$,we get:
$BO - BQ = DO - DP$
$QO = PO$ ... $(3)$.
Now,in quadrilateral $APCQ$,the diagonals $AC$ and $PQ$ bisect each other at $O$ (from equations $(1)$ and $(3)$).
Since the diagonals of a quadrilateral bisect each other,$APCQ$ is a parallelogram.