In the given figure, lines $\mathrm{X}\mathrm{Y}$ and $\mathrm{M}\mathrm{N}$ intersect at $\mathrm{O}.$ If $\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{Y}=90°$ and $a:b=2:3,$ find $c.$

In the given figure, if $x+y=w+z$, then prove that $AOB$ is a line.

In the given figure, $POQ$ is a line. Ray $OR$ is perpendicular to line $PQ\mathit{.}\mathit{ }OS$ is another ray lying between rays $OP$ and $OR\mathit{.}$ Prove that $\mathrm{\angle }ROS=\frac{1}{2}\left(\mathrm{\angle }QOS-\mathrm{\angle }POS\right).$

In the given figure, if $PQ\parallel ST, \mathrm{\angle }PQR=110°$ and $\mathrm{\angle }RST=130°,$ find $\mathrm{\angle }QRS.$

In the given figure, if $AB\parallel CD, \mathrm{\angle }APQ=50°$ and $\mathrm{\angle }PRD=127°,$ find $x$ and $y$.

In the given figure, $PQ$ and $RS$ are two mirrors placed parallel to each other. An incident ray $AB$ strikes the mirror $PQ$ at $B,$ the reflected ray moves along the path $BC$ and strikes the mirror $RS$ at $C$ and again reflects back along $CD.$ Prove that $AB\parallel CD.$