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.

Answer 1

Ray OS stands on the line POQ, so that

∠POS + ∠SOQ = 180°

$x$ + ∠SOQ = 180°

∠SOQ = 180° - $x$

Now, Ray OT bisects ∠SOQ

∠SOT = $\frac{1}{2}$ $\times$ ∠SOQ

∠SOT = $\frac{1}{2}$ $\times$ (180 - $x$)°

∠SOT = $\frac{180°}{2}$ - $\frac{x}{2}$ = $90°$ - $\frac{x}{2}$

Now, Ray OR bisects ∠POS

∠ROS = $\frac{1}{2}$ $\times$ ∠POS

∠ROS = $\frac{1}{2}$ $\times$ $x$ = $\frac{x}{2}$

Now, ∠ROT = ∠SOT + ∠ROS

So, ∠ROT = $90°$ - $\frac{x}{2}$ + $\frac{x}{2}$

⇒ ∠ROT = 90°