In a rectangle $ABCD$,if $AB^{2} + BC^{2} = 64$,then find the length of the diagonal $AC$.

In $\Delta ABC$ and $\Delta DEF$,$\angle A \cong \angle E$ and $m \angle A + m \angle B = m \angle D + m \angle E$. Then the correspondence $ABC \leftrightarrow \ldots$ between them is a similarity.

If $\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow DEF$,and $2 m \angle X = 3 m \angle Y$ with $m \angle Z = 30^{\circ}$,find $m \angle E$. (in $^{\circ}$)

In $\Delta ABC$,$D$ is the midpoint of $\overline{BC}$. $\Delta ABC$ and $\Delta BDE$ are equilateral triangles. Then,the ratio of the areas of $\Delta ABC$ and $\Delta BDE$ is.......

In $\Delta XYZ$,$\overline{XN}$ is a median. If $XY^2 + XZ^2 = 200$ and $XN = 8$,find $YZ$.

Prove that the midpoints of the sides of any convex quadrilateral are the vertices of a parallelogram.