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).

Question

KnowledgeBoat · Accepted Answer

Since, OR ⊥ PQ ∴ ∠ROP = 90° and ∠ROQ = 90° ∴ ∠ROS = 90° - ∠POS .......(1) ⇒ ∠QOS = ∠QOR + ∠ROS ⇒ ∠QOS = 90° + ∠ROS ⇒ 90° = ∠QOS - ∠ROS .....(2) Substituting value of 90° from equation (2) in equation (1), we get : ⇒ ∠ROS = (∠QOS - ∠ROS) - ∠POS ⇒ ∠ROS + ∠ROS = ∠QOS - ∠POS ⇒ 2(∠ROS) = ∠QOS - ∠POS ⇒ ∠ROS = (∠QOS - ∠POS) Hence, proved ∠ROS = (∠QOS - ∠POS).

Mathematics

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).

Lines & Angles

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