Opposite angles of a quadrilateral $ABCD$ are equal. If $AB = 4 \, cm$,determine $CD$ (in $cm$).

$ABCD$ is a parallelogram and $AP$ and $CQ$ are perpendiculars from vertices $A$ and $C$ on diagonal $BD$ (see figure). Show that $AP = CQ$.

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

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

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 figure). Show that $AD \parallel CF$ and $AD = CF$.