5. 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).
Solution: In the question, it is given that (OR ⊥ PQ) and ∠POQ = 180° So, ∠POS+∠ROS+∠ROQ = 180° Now, ∠POS+∠ROS = 180°- 90° (Since ∠POR = ∠ROQ = 90°) ∴ ∠POS + ∠ROS = 90° Now, ∠QOS = ∠ROQ+∠ROS It is given that ∠ROQ = 90°, ∴ ∠QOS = 90° +∠ROS Or, ∠QOS – ∠ROS = 90° As ∠POS + ∠ROS = 90° and ∠QOS – ∠ROS = 90°, we get ∠POS + ∠ROS = ∠QOS – ∠ROS 2 ∠ROS + ∠POS = ∠QOS Or, ∠ROS = ½ (∠QOS – ∠POS) (Hence proved).
