Question

In the adjoining figure, ray OR is perpendicular to the line PQ.
OS is another ray standing on line PQ. Prove that
∠ROS = 1/2(∠QOS - ∠POS)​

Answer 1

━━━━━━━━━━━━━━━━━━━━

✤ Required Proof:

✒ GiveN:

• OR is perpendicular to PQ.
• OS is another line standing on PQ.

✒ To Prove:

• ∠ROS = 1/2( ∠QOS - ∠POS)

━━━━━━━━━━━━━━━━━━━━

✤ How to solve?

First of all, we need to observe the figure carefully because it contains many useful information which help in solving the problem. Some useful informations are ∠POR = ∠QOR = 90°, ∠POQ = 180° etc. So, by using these informations, let's proceed proving the given statement$.$

━━━━━━━━━━━━━━━━━━━━

✤ Solution:

☁️ Refer to the attachment...

• ∠POR = ∠QOR = 90°...........(1)

[Because, OR ⊥ PQ at O]

Now, we have,

➝ ∠QOS = ∠QOR + ∠ ROS

➝ ∠QOS = 90° + ∠ROS......(2) [from (1)]

➝ ∠POS + ∠ROS = ∠POR

➝ ∠POS = ∠POR - ∠ROS

➝ ∠POS = 90° - ∠ROS......(3) [from (1)]

Subtracting (3) from (2),

➝ ∠QOS - ∠POS = (90° + ∠ROS) - (90° - ∠ROS)

➝ ∠QOS - ∠POS = 2 × ∠ROS

➝ 2 × ∠ROS = ∠QOS - ∠POS

➝ ∠ROS = 1/2 (∠QOS - ∠POS)

✒ Hence, proved!

━━━━━━━━━━━━━━━━━━━━