Question

In figure OR perpendicular PQ
prove that: Angle ROS= ¹/2
( Angle QOS- Angle POS)
​

Answer 1

CORRECT QUESTION :

In Fig. 6.17, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that :-

• ∠ROS = ½(∠QOS - ∠POS).

CORRECT SOLUTION :

Given : POQ is a straight line. Two rays OR & OS stands on the same line. Such that, OR ⊥ PQ & ∠ROQ = ∠ROP = 90°.

To Prove : ∠ROS = ½(∠QOS - ∠POS)

Proof : Since, POQ is a straight line.

The sum of all angles formed on the line PQ will be equals to 180°.

⇒ ∠POS + ∠ROS + ∠ROQ = 180°

⇒ ∠POS + ∠ROS + 90° = 180° (∠ROQ = 90°)

⇒ ∠POS + ∠ROS = 90°

⇒ ∠ROS = 90° - ∠POS _____ (Ⅰ)

∠ROS + ∠ROQ = ∠QOS (from the attached figure),

⇒ ∠ROS + 90° = ∠QOS

⇒ ∠ROS = ∠QOS - 90°_____ (Ⅱ)

Adding both the equations (Ⅰ) & (ⅠⅠ),

⇒ ∠ROS + ∠ROS = 90° - ∠POS + ∠QOS - 90°(90° common in L.H.S & R.H.S will be canceled & ∠ROS after adding will become 2∠ROS)

⇒ 2∠ROS = - ∠POS + ∠QOS

⇒ 2∠ROS = (∠QOS - ∠POS)

⇒ ∠ROS = ½(∠QOS - ∠POS)

Hence, Proved ✔.