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

In $\Delta ABC$,$P$ and $Q$ are the midpoints of $AB$ and $AC$ respectively. If $BC + PQ = 21 \text{ cm}$,then find the length of $BC$.

Diagonals of a parallelogram $ABCD$ intersect at $O.$ If $\angle BOC = 90^{\circ}$ and $\angle BDC = 50^{\circ},$ then $\angle OAB$ is (in $^{\circ}$)

$A$ square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.

In parallelogram $PQRS$,$PQ = 8 \, \text{cm}$ and $QR = 5 \, \text{cm}$,then the perimeter of $PQRS = \ldots \ldots \ldots$ (in $, \text{cm}$)

In $\Delta ABC$,$P, Q,$ and $R$ are the midpoints of $AB, BC,$ and $CA$ respectively. Prove that $PBQR, PQCR,$ and $PQRA$ all are parallelograms.