Diagonals of a quadrilateral $ABCD$ bisect each other. If $\angle A = 35^{\circ}$,determine $\angle B$. (in $^{\circ}$)

In parallelogram $PQRS$,the bisector of $\angle P$ intersects $RS$ at $M$ and the bisector of $\angle R$ intersects $PQ$ at $N$. Prove that $PNRM$ is a parallelogram.

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.

$ABCD$ is a parallelogram in which $P$ and $Q$ are mid-points of opposite sides $AB$ and $CD$ respectively. If $AQ$ intersects $DP$ at $S$ and $BQ$ intersects $CP$ at $R$,show that: $APCQ$ is a parallelogram.

In parallelogram $ABCD$,$DM$ is an altitude corresponding to base $AB$. If $AB = 15 \text{ cm}$ and $\text{ar}(ABCD) = 360 \text{ cm}^2$,then $DM = \ldots \text{ cm}$.

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