$ABCD$ is a parallelogram. If its diagonals are equal,then find the value of $\angle ABC$. (in $^{\circ}$)

Prove that the line segment joining the midpoints of the diagonals of a trapezium is parallel to the parallel sides of the trapezium and is equal to half the difference of the lengths of the parallel sides.

State whether the following statement is true or false:
The diagonals of a rhombus bisect each other at right angles.

$P, Q, R$ 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.

In $\Delta ABC$,$P$,$Q$,and $R$ are the midpoints of $AB$,$BC$,and $AC$ respectively. If $AB = 8 \text{ cm}$,$BC = 9 \text{ cm}$,and $AC = 10.6 \text{ cm}$,find the perimeter of $\Delta PQR$ in $\text{cm}$.

Prove that a diagonal of a parallelogram divides it into two congruent triangles.