Question

In the figure below, ray OS stands on a line POQ, Ray OR and ray OT are angle bisectors of $\angle POS$ and $\angle SOQ$ respectivley. If $\angle POS=x,$

• A. ${180}^{\circ }$
• B. ${90}^{\circ }$
• C. ${120}^{\circ }$
• D. ${75}^{\circ }$

Answer 1

The correct option is B ${90}^{\circ }$

Ray OS stands on the line POQ, so, $\angle POS+\angle SOQ={180}^{\circ }$ (Linear pair)
Given, $\angle POS=x$
Hence, $x+\angle SOQ={180}^{\circ }$
$\Rightarrow \angle SOQ={180}^{\circ }-x$
Now , ray OR bisects $\angle POS$
Hence, $\angle ROS=\frac{1}{2}\times \angle POS.$
$\Rightarrow \frac{1}{2}\times x=\frac{x}{2}$
Similarly, $\angle SOT=\frac{1}{2}\times \angle SOQ$
$\Rightarrow \frac{1}{2}\times ({180}^{\circ }-x)={90}^{\circ }-\frac{x}{2}$
Hence, $\angle ROT=\angle ROS+\angle SOT$
$\Rightarrow \frac{x}{2}+{90}^{\circ }-\frac{x}{2}={90}^{\circ }$