Question

If ray $OS$ stand on a line $POQ$. Ray $OR$ and ray $OT$ are angle bisectors of $\angle POS$ and $\angle SOQ$ respectively. If $\angle POS=x$, find $\angle ROT$.

Answer 1

Given that:-

$\angle POS=x$

Ray OR and ray OT are angle bisectors of $\angle POS$ and $\angle SOQ$ respectively.

$\therefore \angle ROP=\angle ROS=\frac{1}{2}\angle POS..........\left(1\right)$

$\angle SOT=\angle TOQ=\frac{1}{2}\angle SOQ..........\left(2\right)$

From figure,

$\angle POS+\angle SOQ=180°\left(\text{Linear pair}\right)$

From $\left(1\right)\&\left(2\right)$, we have

$2\angle ROS+2\angle SOT=180°$

$\Rightarrow 2(\angle ROS+\angle SOT)=180°$

$\because \angle ROT=\angle ROS+\angle SOT\left(\text{From fig.}\right)$

$\therefore 2\angle ROT=180°$

$\Rightarrow \angle ROT=90°$

Hence, $\angle ROT=90°$.