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 =(∠ QOS –∠ POS).

Question

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 &#x2220; ROS = ( &#x2220; QOS &#x2013; &#x2220; POS).

Philoid · Accepted Answer

It is given in the question that:

OR is perpendicular to line PQ

To prove,

∠ROS = (∠QOS - ∠POS)

Now, according to the question,

∠POR = ∠ROQ = 90^O (Perpendicular)

∠QOS = ∠ROQ + ∠ROS = 90^O + ∠ROS (i)

∠POS = ∠POR - ∠ROS = 90^O - ∠ROS (ii)

Subtracting (ii) from (i), we get

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

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

∠QOS - ∠POS = 2∠ROS

∠ROS = (∠QOS - ∠POS)

Hence, proved