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

Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square.

$A$ diagonal of a parallelogram bisects one of its angles. Prove that it will bisect its opposite angle also.

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

$P, Q, R$ and $S$ are respectively the mid-points of the sides $AB, BC, CD$ and $DA$ of a quadrilateral $ABCD$ such that $AC \perp BD$. Prove that $PQRS$ is a rectangle.

In $\Delta ABC$,$P$,$Q$,and $R$ are the midpoints of the sides. If the perimeter of $\Delta ABC$ is $23 \, cm$,then find the perimeter of $\Delta PQR$ in $cm$.