In 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: OR is perpendicular to line PQ

To prove: ∠ROS = 1/2 (∠QOS - ∠POS)

Proof:

Now, according to the question,

∠POR = ∠ROQ = 90° ( ∵ OR is perpendicular to line PQ)

∠QOS = ∠ROQ + ∠ROS = 90° + ∠ROS ............eq(i)

We can write,

∠POS = ∠POR - ∠ROS = 90° - ∠ROS ...............eq(ii)

Subtracting (ii) from (i), we get

∠QOS - ∠POS = 90^o + ∠ROS – (90° - ∠ROS)

∠QOS - ∠POS = 90^o + ∠ROS – 90° + ∠ROS

∠QOS - ∠POS = 2∠ROS

∠ROS = 1/2 (∠QOS - ∠POS)

Hence, Proved.