In the given figure,if $AB \parallel CD$,$\angle PCD = 130^{\circ}$ and $\angle PBA = 140^{\circ}$,then find $\angle BPC$. (in $^{\circ}$)

In the figure,$m$ and $n$ are two plane mirrors perpendicular to each other. Show that the incident ray $CA$ is parallel to the reflected ray $BD$.

In the given figure,if $PQ \parallel RS$ and $RS \parallel TU$,and $y: z = 7: 8$,find $x$.

Bisectors of angles $B$ and $C$ of a triangle $ABC$ intersect each other at the point $O$. Prove that $\angle BOC = 90^{\circ} + \frac{1}{2} \angle A$.

$\angle ACD$ is an exterior angle of $\Delta ABC$. The bisector of $\angle ABC$ and exterior $\angle ACD$ intersect each other at point $E$. Prove that,$\angle BEC = \frac{1}{2} \angle BAC$.

In the figure,if $AB \parallel CD \parallel EF$,$PQ \parallel RS$,$\angle RQD = 25^{\circ}$ and $\angle CQP = 60^{\circ}$,then $\angle QRS$ is equal to: (in $^{\circ}$)