Question

In the figure, 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

Ray OR stands on a line POQ forming $\angle POS$ and $\angle QOS$

OR and OT the angle bisects of $\angle POS$ and $\angle QOS$ respectively. $\angle POS=x$
But $\angle POS+\angle QOS={180}^{\circ }$ (Linear pair)
$\Rightarrow x+\angle QOS={180}^{\circ }\Rightarrow \angle QOS={180}^{\circ }-x$
$\because$ OR and OT are the bisectors of angle
$\therefore \angle ROS=\frac{x}{2}$
and $\angle TOS=\frac{{180}^{\circ }-x}{2}$
$\therefore \angle ROT=\angle ROS+\angle TOS=\frac{x}{2}+\frac{{180}^{\circ }-x}{2}=\frac{x+{180}^{\circ }-x}{2}=\frac{{180}^{\circ }}{2}={90}^{\circ }$