In the given figure is a line. Ray is perpendicular to line is another ray lying between rays and

Prove that

Given that, is a line and is perpendicular to line .

⇒ ∠POR = 90°

The sum of linear pair is always equal to 180°

∴ ∠POS + ∠ROS + ∠POR = 180°

Substitute ∠POR = 90°

⇒90° + ∠POS + ∠ROS = 180°

⇒∠POS + ∠ROS = 90°

∴ ∠ROS = 90°-∠POS----(1)

⇒ ∠QOR = 90°

Given that OS is another ray lying between OP and OR

⇒∠QOS-∠ROS = 90°

∴∠ROS = ∠QOS-90°----(2)

On adding two equations (1) and (2) we get

2∠ROS = ∠QOS-∠POS

⇒ ∠ROS = (∠QOS-∠POS)

So, ∠ROS = (∠QOS-∠POS) is proved.