The figure formed by joining the mid-points of the sides of a quadrilateral $ABCD,$ taken in order,is a square only if,

$PQ$ and $RS$ are two equal and parallel line segments. Any point $M$ not lying on $PQ$ or $RS$ is joined to $Q$ and $S$. Lines are drawn through $P$ parallel to $QM$ and through $R$ parallel to $SM$,meeting at $N$. Prove that line segments $MN$ and $PQ$ are equal and parallel to each other.

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral $PQRS,$ taken in order,is a rectangle,if

In parallelogram $ABCD$,$\angle A = \angle B - 30^{\circ}$,then find $\angle A$. (in $^{\circ}$)

In parallelogram $ABCD$,$\angle A = \angle B + 50^{\circ}$. Find the measure of each angle of the quadrilateral $ABCD$.

In $\Delta PQR$,$A$,$B$,and $C$ are the mid-points of $PQ$,$QR$,and $RP$ respectively. If the perimeter of $\Delta ABC$ is $18.6 \, cm$,then the perimeter of $\Delta PQR$ is $\ldots \ldots \ldots cm$.