In a rhombus $ABCD,$ the altitude from $D$ to the side $AB$ bisects $AB$. Find the angles of the rhombus.

In a parallelogram $ABCD$, points $M$ and $N$ have been taken on opposite sides $AB$ and $CD$ respectively such that $AM=CN$. Show that $AC$ and $MN$ bisect each other.

In the adjoining figure, $ABCD$ is a parallelogram. If $P$ and $Q$ are points on $AD$ and $BC$ respectively such that $AP=\frac{1}{3}AD$ and $CQ=\frac{1}{3}BC$, prove that $AQCP$ is a parallelogram.

In the adjoining figure, $ABCD$ is a parallelogram whose diagonals intersect each other at $O.$ A line segment $EOF$ is drawn to meet $AB$ at $E$ and $DC$ at $F.$ Prove that $OE=OF$.

In the adjoining figure, $ABCD$ is a parallelogram in which $AB$ is produced to $E$ so that $BE=AB$. Prove that $ED$ bisects $BC$.