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

Academic Mathematics NCERT Class 9

Given:

$POQ$ is a line, Ray $OR$ is perpendicular to line $PQ$ and $OS$ 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 $OR \perp POQ$.

This implies,

$\angle POR = 90^o$

$\angle POS + \angle ROS = 90^o$.....…(i)

$\angle ROS = 90^o - \angle POS$

$\angle POS + \angle QOS = 180^o$ (Linear pair)

$= 2(∠POS + ∠ROS)$ [From (i)]

$\angle POS + \angle QOS = 2\angle ROS + 2\angle POS$

$2\angle ROS = \angle POS + \angle QOS - 2\angle POS$

$2\angle ROS =\angle QOS - \angle POS$

$\angle ROS = \frac{1}{2}(\angle QOS - \angle POS)$

Hence proved.

Updated on: 10-Oct-2022

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