Intersecting Lines and Non-intersecting Lines

It is given that $\mathrm{\angle }XYZ=64°$ and $XY$ is produced to point $P$. Draw a figure from the given information. If ray $YQ$ bisects $\mathrm{\angle }ZYP,$ find $\mathrm{\angle }XYQ$ and reflex $\mathrm{\angle }QYP.$

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 $x+y=w+z$, then prove that $AOB$ is a line.

In the given figure, $\mathrm{\angle }PQR=\mathrm{\angle }PRQ,$ then prove that $\mathrm{\angle }PQS=\mathrm{\angle }PRT$.

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, lines $\mathrm{A}\mathrm{B}$ and $\mathrm{C}\mathrm{D}$ intersect at $\mathrm{O}.$ If $\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{C}+\mathrm{\angle }\mathrm{B}\mathrm{O}\mathrm{E}=70°$ and $\mathrm{\angle }\mathrm{B}\mathrm{O}\mathrm{D}=40°$, find $\mathrm{\angle }\mathrm{B}\mathrm{O}\mathrm{E}$ and reflex $\mathrm{\angle }\mathrm{C}\mathrm{O}\mathrm{E}$.