Question

"Question 5 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

Class 9 - Math - Lines and Angles Page 97"

Answer 1

Hi friend,

It is given that OR is perpendicular to PQSo that ∠POR = 90°sum of angle in linear pair always equal to 180°∠POS + ∠SOR + ∠POR = 180°Plug ∠POR = 90°90°+∠SOR + ∠POR = 180°∠SOR + ∠POR = 90°∠ROS = 90° − ∠POS … (1)∠QOR = 90°Given that OS is another ray lying between raysOP and OR so that∠QOS − ∠ROS = 90°∠ROS = ∠QOS − 90° … (2)On adding equations (1) and (2), we obtain2 ∠ROS = ∠QOS − ∠POS∠ROS = 1/2(∠QOS − ∠POS)

Hope is helped ya.

Answer 2

Given,
OR is perpendicular to line PQ

To prove,
∠ROS = 1/2(∠QOS – ∠POS)

Proof:

∠POR =∠QOR = 90° (Perpendicular)

Now, ∠POR=90°

∠POS + ∠ROS=90°=∠QOR

 ∠POS + ∠ROS=∠QOR

On adding ∠ROS both sides,

∠POS + ∠ROS+∠ROS =∠QOR+∠ROS

 

∠POS+ 2∠ROS= ∠QOS

[∠QOS=∠QOR+∠ROS]

 

2∠ROS= ∠QOS - ∠POS

 

∠ROS= ½[∠QOS - ∠POS]

 

Hence, ∠ROS= ½[∠QOS - ∠POS]

 

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Hope this will help you