If $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$,$2 AB = PQ$,and $BC = 10$,then $QR = \dots$

The lengths of the sides of a triangle are $m^{2}+n^{2}$,$2mn$,and $m^{2}-n^{2}$,where $m > n > 0$. Prove that the triangle is a right-angled triangle.

In rhombus $ABCD$,$AC = 18$ and $BD = 24$. Find the perimeter of rhombus $ABCD$.

$\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow EFD$. $AB : BC : CA = 4 : 3 : 5$. If the perimeter of $\Delta DEF$ is $36$,find the measures of all the sides of $\Delta DEF$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $D$ is a point on $\overline{BC}$. Prove that $AD^{2} + BC^{2} = AC^{2} + BD^{2}$.

Is the following statement true? Why?
"Two quadrilaterals are similar,if their corresponding angles are equal".