In the figure,$ABCD$ and $AEFG$ are two parallelograms. If $\angle C = 55^{\circ},$ determine $\angle F.$ (in $^{\circ}$)

In a quadrilateral $ABCD$,$\angle A : \angle B : \angle C : \angle D = 4 : 5 : 4 : 5$. Find the measure of each angle of the quadrilateral and state the type of quadrilateral $ABCD$.

In $\triangle ABC$,$AB = 5\, cm$,$BC = 8\, cm$,and $CA = 7\, cm$. If $D$ and $E$ are respectively the mid-points of $AB$ and $BC$,determine the length of $DE$. (in $, cm$)

$(1)$ $A$ quadrilateral in which only one pair of opposite sides are parallel is a $\ldots \ldots \ldots$.
$(2)$ The diagonals of a rectangle are $\ldots \ldots \ldots$.

In parallelogram $PQRS$,diagonal $PR$ bisects $\angle P$. Prove that diagonal $PR$ bisects $\angle R$ also and $PQRS$ is a rhombus.

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