In the given figure,$POQ$ is a straight line. The value of $x$ is: (in $^{\circ}$)

In $\Delta ABC$,the bisectors of $\angle B$ and $\angle C$ intersect each other at $I$. Prove that $\angle BIC = 90^{\circ} + \frac{1}{2} \angle A$.

In the given figure,if $AB \parallel CD$,$\angle AXY = 80^{\circ}$ and $\angle XZD = 140^{\circ}$,then find $x$ and $y$.

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 following figure,ray $BA$ is perpendicular to line $CD$. If $x: y: z = 4: 5: 6$,then find the values of $x, y,$ and $z$.

In the figure,$DE \parallel QR$ and $AP$ and $BP$ are bisectors of $\angle EAB$ and $\angle RBA$ respectively. Find $\angle APB.$ (in $^{\circ}$)