Question

In the below fig, POQ is a line. Ray OR is perpendicular to line OS is another ray lying

between rays OP and OR. Prove that ∠ROS = 1 (∠QOS − ∠POS).

Answer 1

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°             .........(2)

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

2`∠`ROS = `∠`QOS - `∠`POS

`∠`ROS = `1/2` (`∠`QOS - `∠`POS )