Prove that diagonals of a rectangle are of equal length.

Prove that diagonals of a square are equal and they bisect each other at right angles.

$X$ and $Y$ are the midpoints of opposite sides $AB$ and $CD$ of parallelogram $ABCD$. Prove that $AXCY$ is a parallelogram.

In the adjacent figure, $AB$ and $DC$ are two parallel lines which are intersected by transversal $l$ at points $P$ and $R$respectively. Prove that the bisectors of the interior angles make a rectangle.

$ABC$ is an isosceles triangle in which $AB=AC$, $AD$ bisects the exterior angle $PAC$ and $CD\parallel BA$. Prove that $\angle DAC=\angle BCA$

$ABC$ is an isosceles triangle in which $AB=AC$, $AD$ bisects the exterior angle $PAC$ and $CD\parallel BA$. Prove that quadrilateral $ABCD$ is a parallelogram.

$ABCD$ is a parallelogram and $BD$ is one of its diagonals. $AP$ and $CQ$ are perpendiculars on $BD$ from the vertices $A$ and $C$ respectively. Prove that $△APB\cong △CQD$

$ABCD$ is a parallelogram and $BD$ is one of its diagonals. $AP$ and $CQ$ are perpendiculars on $BD$ from the vertices $A$ and $C$ respectively. Prove that $AP=CQ$.

$ABCD$ is a rectangle in which diagonal $AC$ bisects both the angles $A$ and $C$. Then prove that $ABCD$ is a square.