Question

In the figure below, ray OS stands on a line POQ. Ray OR and ray OT are angle bisectors of ∠POS and ∠SOQ, respectively. If ∠POS = x, find ∠ROT.

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

Answer 1

The correct option is C

${90}^{\circ }$

Ray OS stands on the line POQ, so that,

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

But, $\angle$$POS=x$

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

$\angle$$SOQ=$ ${180}^{\circ }$$–x$

Now, ray OR bisects POS.

Hence, $\angle$$ROS=$ $\frac{1}{2}$ $\times$ $\angle$$POS$ => $\frac{1}{2}$ $\times$ x = $\frac{x}{2}$

Similarly, $SOT=$$\frac{1}{2}$ $\times$ $\angle$$SOQ$ => $\frac{1}{2}$ $\times$ ( ${180}^{\circ }$ - x) = ${90}^{\circ }$ – $\frac{x}{2}$

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