In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that
$ \angle ROS =\frac{1}{2} (\angle QOS -\angle POS)$
"

Given :

POQ is a line.

Ray OR is perpendicular to line PQ.

OS is another ray lying between rays OP and OR.

To do :

We have to prove that $ \angle ROS =\frac{1}{2} (\angle QOS -\angle POS)$

Solution :

Ray OR is perpendicular to line PQ.

This implies,

$∠ROP = 90°$ and $∠ROQ = 90°$

Therefore,

$∠ROP = ∠ROQ$

$∠POS + ∠ROS = ∠ROQ$

$∠POS + ∠ROS = ∠QOS - ∠ROS$

$∠SOR + ∠ROS = ∠QOS - ∠POS$

$2(∠ROS) = ∠QOS - ∠POS$

$∠ROS = \frac{1}{2} (∠QOS - ∠POS)$

Hence proved.

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Updated on: 10-Oct-2022

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