In parallelogram $PQRS$,$PQ = 8 \, \text{cm}$ and $QR = 5 \, \text{cm}$,then the perimeter of $PQRS = \ldots \ldots \ldots$ (in $, \text{cm}$)

$ABCD$ is a parallelogram and $P$ and $Q$ are points on the diagonal $AC$ such that $AP = PQ = QC$. Prove that $BQ \parallel DP$ and $BD$ bisects $PQ$.

Diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect each other at $O$ such that $OA : OC = 3 : 2$. Is $ABCD$ a parallelogram? Why or why not?

If the diagonals of a parallelogram are equal,then show that it is a rectangle.

In $\Delta ABC$ and $\Delta DEF$,$AB = DE$,$AB \parallel DE$,$BC = EF$ and $BC \parallel EF$. Vertices $A, B$ and $C$ are joined to vertices $D, E$ and $F$ respectively (see Fig). Show that quadrilateral $ACFD$ is a parallelogram.

In parallelogram $ABCD$,$\angle A$ is an obtuse angle,$AM \perp BC$,and $AN \perp CD$. If $\angle MAN = 60^{\circ}$,find each of the angles of quadrilateral $ABCD$.