In the figure, 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).

Given, OR is perpendicular to line PQ
To prove: ∠ROS = 1/2(∠QOS - ∠POS)
According to the question,
∠POR = ∠ROQ = 90° ∠Perpendicular
∠QOS = ∠ROQ + ∠ROS = 90° + ∠ROS — (i)
∠POS = ∠POR - ∠ROS = 90° - ∠ROS — (ii)
Subtracting (ii) from (i)
∠QOS - ∠POS = 90° + ∠ROS - (90° - ∠ROS)
→ ∠QOS - ∠POS = 90° + ∠ROS + 90° + ∠ROS
→ ∠QOS - ∠POS = 2∠ROS
→ ∠ROS = 1/2(∠QOS - ∠POS)
[hence proved]