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 the given figure,if $\angle D = \angle C$,then is it true that $\triangle ADE \sim \triangle ACB$? Why?

In $\Delta ABC$,$\overline{AD}$ is a median and $AB^2 + AC^2 = 148$. If $AD = 7$,then $BC = \ldots$

Prove that the area of the equilateral triangle drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides of the triangle.

In $\Delta XYZ$,$m\angle Y = 90^{\circ}$ and $\overline{YM}$ is an altitude to the hypotenuse $\overline{XZ}$. If $YM = 12$ and $XM = 8$,find $XZ$.

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $\frac{AB}{5} = \frac{AC}{3}$ and $BD = 4.5$,find $BC$.