$XYZW$ is a rhombus. If the diagonals $XZ$ and $YW$ intersect at $P$,then $\angle XPY = \ldots \ldots \ldots$ (in $^o$)

$ABCD$ is a trapezium in which $AB \parallel CD$ and $AD = BC$ (see Fig). Show that diagonal $AC =$ diagonal $BD$.

In parallelogram $ABCD$,$AB = 12 \, cm$. Altitudes $DM$ and $DN$ correspond to bases $AB$ and $BC$ respectively. If $DM = 5 \, cm$ and $DN = 6 \, cm$,then find the length of $BC$ in $cm$.

$ABCD$ is a rectangle in which diagonal $AC$ bisects $\angle A$ as well as $\angle C$. Show that: diagonal $BD$ bisects $\angle B$ as well as $\angle D$.

Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles,then it is a square.

In a parallelogram $ABCD$,$\angle A = 5 \angle B$,then $\angle C = \ldots$ (in $^o$)