In $\square PQRS$,$\overline{PQ} \parallel \overline{RS}$ and $\overline{PR} \cap \overline{QS} = \{X\}$. If $XP = 2x + 4$,$XQ = 4x - 2$,$XR = x + 1$,and $XS = 4$,find the value of $x$.

In $\Delta ABC$,$\Delta PQR$,and $\Delta XYZ$,the correspondences $ABC \leftrightarrow QPR$ and $PQR \leftrightarrow YZX$ are similarities. Then,the correspondence $ABC \leftrightarrow \dots$ is a similarity.

If $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow QRP$ and $m \angle A : m \angle B : m \angle C = 2 : 3 : 5$,then $m \angle R = \dots$ (in $^{\circ}$)

Given $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $2 m \angle P = 3 m \angle Q$ and $m \angle C = 100^{\circ}$,find $m \angle B$. (in $^{\circ}$)

In the given figure,$BD$ and $CE$ intersect each other at the point $P$. Is $\triangle PBC \sim \triangle PDE$ ? Why?

$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $AB = 8$,$PQ = 14$ and the perimeter of $\Delta ABC$ is $20$,find the perimeter of $\Delta PQR$.