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

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

In $\Delta XYZ$,$A$ and $B$ are the midpoints of $XY$ and $XZ$ respectively. If $AB = 7.5 \, cm$,then find the length of $YZ$ in $cm$.

$ABCD$ is a trapezium such that $AB || CD$. If $\angle A = y + 60^{\circ}$,$\angle B = x + 60^{\circ}$,$\angle C = 3x - 40^{\circ}$ and $\angle D = 3y - 80^{\circ}$,then find the measure of each angle of $ABCD$.

Points $P$ and $Q$ have been taken on opposite sides $AB$ and $CD$ respectively of a parallelogram $ABCD$ such that $AP = CQ$ (see figure). Show that $AC$ and $PQ$ bisect each other.

$E$ is the mid-point of the side $AD$ of the trapezium $ABCD$ with $AB \parallel DC$. $A$ line through $E$ drawn parallel to $AB$ intersects $BC$ at $F$. Show that $F$ is the mid-point of $BC$.