For the correspondence $ABC \leftrightarrow RPQ$ between $\Delta ABC$ and $\Delta PQR$,$\angle B$ corresponds to $\ldots \ldots \ldots \ldots$

In quadrilateral $ABCD$,$P$ is the midpoint of $\overline{BC}$. If $\overline{DP}$ produced intersects $\overline{AB}$ produced at $Q$,prove that $AB = 2CD$ given that $ABCD$ is a parallelogram.

In rhombus $ABCD$,$AC = 15$ and $BD = 36$. Find the perimeter of rhombus $ABCD$.

In $\Delta PQR$ and $\Delta XYZ$,if $\frac{PQ}{XY} = \frac{QR}{YZ} = \frac{PR}{XZ} = \frac{2}{5}$,then the ratio of their areas $\text{Area}(\Delta PQR) : \text{Area}(\Delta XYZ) = \ldots$

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

In $\Delta ABC,$ the midpoints of the sides are $P, Q, R.$ In $\Delta PQR,$ the midpoints of the sides are $X, Y, Z.$ If the area of $\Delta XYZ$ is $20,$ find the area of $\Delta PQR$ and $\Delta ABC.$