$\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow FDE$. If $m \angle E + m \angle F = 130^{\circ}$,then $m \angle B = \dots$ (in $^{\circ}$)

Given $\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XYZ$. If $\frac{PQ}{3} = \frac{XY}{5}$ and $QR = 9$,find the length of $YZ$.

$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. $\overline{AD} \perp \overline{BC}$,$D \in \overline{BC}$ and $\overline{XM} \perp \overline{YZ}$,$M \in \overline{YZ}$. Prove that $\frac{AD}{XM} = \frac{BC}{YZ}$.

In parallelogram $ABCD$,$A-P-B$ and $AP = \frac{2}{3} AB$. $\overline{DP}$ intersects $\overline{AC}$ at $Q$. Find the ratio of the areas of $\Delta APQ$ and $\Delta CDQ$.

It is given that $\triangle ABC \sim \triangle EDF$ such that $AB = 5 \, cm$,$AC = 7 \, cm$,$DF = 15 \, cm$,and $DE = 12 \, cm$. Find the lengths of the remaining sides of the triangles.

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.......