In parallelogram $ABCD$,$\angle A = \angle B - 30^{\circ}$,then find $\angle A$. (in $^{\circ}$)

In parallelogram $ABCD$,two points $P$ and $Q$ are taken on diagonal $BD$ such that $DP = BQ$ (see Fig). Show that: $AP = CQ$.

Which of the following is not true for a parallelogram?

$ABC$ is an isosceles triangle in which $AB = AC$. $AD$ bisects exterior angle $PAC$ and $CD \parallel AB$ (see Fig). Show that $ABCD$ is a parallelogram.

$XYZW$ 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 $\Delta ABC \cong \Delta BAD$.