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In the figure, POQ is a line. Ray OR is perpendicular to line

Question

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

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Michael Discussed

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

∠ ROS = \frac{1}{2} (∠ QOS - ∠ POS)

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Text solutionVerified

Given,

OR

is perpendicular to line

PQ

To prove:

∠ ROS = \frac{1}{2} (∠ QOS -∠ POS)

According to the question,

∠ POR = ∠ ROQ = 9 0^{\circ}

∣

Perpendicular

∠ QOS = ∠ ROQ + ∠ ROS = 9 0^{\circ} + ∠ ROS

--- (i)

∠ POS = ∠ POR - ∠ ROS = 9 0^{\circ} - ∠ ROS

--- (ii)

Subtracting (ii) from (i)

∠ QOS - ∠ POS = 9 0^{\circ} + ∠ ROS - (9 0^{\circ} - ∠ ROS)

\Rightarrow ∠ QOS - ∠ POS = 9 0^{\circ} + ∠ ROS - 9 0^{\circ} + ∠ ROS

\Rightarrow ∠ QOS - ∠ POS = 2∠ ROS

\Rightarrow ∠ ROS = \frac{1}{2} (∠ QOS -∠ POS)

[h e n ce p ro v e d]

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Question Text	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 $∠ ROS = \frac{1}{2} (∠ QOS - ∠ POS)$ .
Answer Type	Text solution:1
Upvotes	150