Question

In Fig. 8.48, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that ∠ROS = 1/2 (∠QOS –∠POS.)

Answer 1

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]

Hope this answer will help you…..

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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°

 ∠POS + ∠ROS = ∠QOR

adding ∠ROS both sides,

=>∠POS + ∠ROS + ∠ROS = ∠QOR +∠ROS

=>∠POS + 2∠ROS = ∠QOS [∠QOS =∠QOR + ∠ROS]

=>2∠ROS = ∠QOS - ∠POS

∠ROS = ½[∠QOS - ∠POS]