#### 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).

#### Solution

Given that, OR perpendicular

∴`∠`*P**O**R *= 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 *)

Is there an error in this question or solution?

Solution 19. In the Below 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 (∠Qos − ∠Pos). Concept: Concept to Lines and Angles.