In the below fig. 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 that

OR perpendicular

∴ ∠POR = 90°

∠POS + ∠SOR = 90         [∴ ∠POR = ∠POS + ∠SOR]

∠ROS = 90° − ∠POS  ... (1)

∠QOR = 90 (∵ OR ⊥ PQ)

∠QOS − ∠ROS = 90°

∠ROS = ∠QOS − 90°

By adding (1) and (2) equations, we get

∴ ∠ROS = ∠QOS − ∠POS

∠ROS = 1/2(∠QOS − ∠POS)