Question

In Figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that \(∠\)ROS = \(\frac{1}{2}\) (\(∠\)QOS – \(∠\)POS)

Answer 1

It is given that (OR ⊥ PQ) and \(∠\)POQ = 180°

We can rewrite it as \(∠\)ROP = \(∠\)ROQ = 90⁰
Since, \(∠\)ROP =\(∠\)ROQ
It can be written as
\(∠\)POS + \(∠\)ROS = \(∠\)ROQ
\(∠\)POS + \(∠\)ROS = \(∠\)QOS – \(∠\)ROS
\(∠\)SOR + \(∠\)ROS = \(∠\)QOS – \(∠\)POS

\(⇒\) 2\(∠\)ROS = \(∠\)QOS – \(∠\)POS
i.e, \(∠\)ROS = \(\frac{1}{2}\) (\(∠\)QOS – \(∠\)POS)