Diagonals of a parallelogram $ABCD$ intersect at $O.$ If $\angle BOC = 90^{\circ}$ and $\angle BDC = 50^{\circ},$ then $\angle OAB$ is (in $^{\circ}$)

$PQRS$ is a parallelogram. If the two diagonals are equal,find the measure of $\angle PQR$. (in $^{\circ}$)

In parallelogram $ABCD$,the diagonals intersect at $P$. If $PA = 3.8 \, cm$ and $PB = 5.2 \, cm$,then $BD = \dots \dots \dots cm$.

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 $AC = DF$.

Show that the diagonals of a square are equal and bisect each other at right angles.

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