One angle of a quadrilateral is $108^{\circ}$ and the remaining three angles are equal. Find each of the three equal angles. (in $^{\circ}$)

In $\Delta ABC$,$AD$ is a median and $E$ is the midpoint of $AD$. $BE$ is extended to intersect $AC$ at $F$. Prove that $AF = \frac{1}{3} AC$.

In a rhombus $ABCD$,$\angle B = 80^{\circ}$,then $\angle ADB = \ldots$ (in $^{\circ}$)

In quadrilateral $ABCD$,$\angle A : \angle B : \angle C : \angle D = 1 : 4 : 2 : 2$. Find the measure of each angle of the quadrilateral.

$P, Q, R$ 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.

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 $32 \, \text{cm}$.