In $\Delta PQR$,the bisector of $\angle P$ intersects $\overline{QR}$ at $S$. If $PQ = 15$,$QS = 10$,and $SR = 8$,then $PR = $.........

If $\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow DEF$,and $2 m \angle X = 3 m \angle Y$ with $m \angle Z = 30^{\circ}$,find $m \angle E$. (in $^{\circ}$)

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. Then,$BM$ is the geometric mean of..............

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $AB = 8$,$AC = 10$ and $BD = 3.2$,then $DC = \ldots$

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x+1$,$MB = 2x-2$,$AN = 2x+2$ and $NC = 3x+1$,find the value of $x$.

In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$,$M \in \overline{AD}$ and $N \in \overline{BC}$. If $\overline{MN} \parallel \overline{AB}$,prove that $\frac{DM}{MA} = \frac{CN}{NB}$.