$XYZW$ is a square. If $XY = 17 \text{ cm}$,then find the area of $XYZW$ in $\text{cm}^2$.

$ABCD$ is a quadrilateral in which $AB \parallel DC$ and $AD = BC$. Prove that $\angle A = \angle B$ and $\angle C = \angle D$.

$ABCD$ is a rectangle in which diagonal $BD$ bisects $\angle B$. Show that $ABCD$ is a square.

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

The area of a trapezium is $475 \, cm^2$ and the height is $19 \, cm$. Find the lengths of its two parallel sides if one side is $4 \, cm$ greater than the other.

$A$ 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.