$ABCD$ is a parallelogram. If $\operatorname{ar}(ABC) = 42 \, \text{cm}^2$,then find $\operatorname{ar}(ABCD)$ in $\text{cm}^2$.

$ABCD$ is a rhombus. Show that diagonal $AC$ bisects $\angle A$ as well as $\angle C$ and diagonal $BD$ bisects $\angle B$ as well as $\angle D$.

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

$ABCD$ is a parallelogram. If its diagonals are equal,then find the value of $\angle ABC$. (in $^{\circ}$)

Two parallel lines $l$ and $m$ are intersected by a transversal $p$ (see Fig). Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.

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 $\Delta ABC \cong \Delta DEF$.