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 $^{\circ}$)

In $\Delta PQR$,$\angle P$ is a right angle and $\overline{PX}$ is an altitude. If $QX = 4$ and $RX = 5$,then $PQ = \ldots \ldots$

$\square ABCD$ is a rectangle. If $AB^{2} + BC^{2} + CD^{2} + DA^{2} = 128$,find the length of diagonal $\overline{AC}$.

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB = 8$,$BC = 18$,and $AD = 7$,then $AC = \ldots$

In $\Delta ABC$ and $\Delta DEF$,$\frac{AB}{DF} = \frac{BC}{EF}$ and $\angle B \cong \angle F$. Then,by $\ldots \ldots \ldots$ condition,the correspondence $ABC \leftrightarrow DFE$ is a similarity.

In the figure,if $\angle A = \angle C$,$AB = 6 \, cm$,$BP = 15 \, cm$,$AP = 12 \, cm$,and $CP = 4 \, cm$,then find the lengths of $PD$ and $CD$ (in $cm$).