The perimeter of rectangle $ABCD$ is $112 \, cm$ and $AB : BC = 5 : 3$. Find the length of $AB$ in $cm$.

Points $P$ and $Q$ have been taken on opposite sides $AB$ and $CD$ respectively of a parallelogram $ABCD$ such that $AP = CQ$ (see figure). Show that $AC$ and $PQ$ bisect each other.

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

State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a rhombus. If $AB = 12 \, \text{cm}$, then the perimeter of $ABCD$ is $48 \, \text{cm}$.
$(2)$ $PQRS$ is a square. If $PR = 8 \, \text{cm}$, then the perimeter of $PQRS$ is $32 \, \text{cm}$.

$(1)$ The diagonals of a rhombus are ....... to each other.
$(2)$ The diagonals of a parallelogram ...... each other.

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