In a rhombus $ABCD$,if $AB = 13 \text{ cm}$,then the perimeter of $ABCD = \ldots \ldots \ldots \text{ cm}$.

$P, Q, R$ and $S$ are respectively the mid-points of the sides $AB, BC, CD$ and $DA$ of a quadrilateral $ABCD$ such that $AC \perp BD$. Prove that $PQRS$ is a rectangle.

$ABCD$ is a rectangle in which diagonal $BD$ bisects $\angle B$. Show that $ABCD$ is a square.

In quadrilateral $ABCD$,$\angle A + \angle D = 180^{\circ}$. What special name can be given to this quadrilateral?

In rhombus $PQRS$,$PR = 40 \, cm$ and $QS = 42 \, cm$,then $PQ = \dots \dots \dots \, cm$.

$ABCD$ 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 $PQ$.