Question

In the following figure, ray OS stands on a line POQ, ray OR and ray OT are angle bisectors of $\angle$POS and $\angle$QOS respectively.
If $\angle$POS = x, find $\angle$ROT.

Answer 1

Given: POQ is a straight line and ray OR and ray OT are angle bisectors of $\angle$POS and $\angle$QOS respectively.
Angles $\angle POS\mathrm{and}\angle QOS$ form a linear pair, so
$\angle POS+\angle SOQ={180}^{\circ }$
Since OR and OT are angle bisectors of $\angle$POS and $\angle$QOS respectively, therefore
$$\begin{gathered} 2\angle ROS+\angle SOQ={180}^{\circ } \\ \Rightarrow 2\angle ROS+2\angle SOT={180}^{\circ } \\ \Rightarrow \angle ROS+\angle SOT={90}^{\circ } \\ \Rightarrow \angle ROT={90}^{\circ } \end{gathered}$$
Hence, $\angle ROT={90}^{\circ }$ which is independent of x.