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 respectively. If $\angle$ POS = $x$, then find $\angle$ ROT.

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

Answer 1

The correct option is C

${90}^{\circ }$

Ray OS stands on the line POQ, so,

$\angle POS+\angle SOQ={180}^{\circ }$ {Linear pair}

Given, $\angle POS=x$.

$\text{Hence},x+\angle SOQ={180}^{\circ }$

$\Rightarrow \angle SOQ={180}^{\circ }–x$.

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 , $\angle ROT=\angle ROS+\angle SOT$
$\Rightarrow \frac{x}{2}+{90}^{\circ }–\frac{x}{2}={90}^{\circ }$.