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)