Question

POQ is a line . Ray OR is perpendicular to line PQ . OS is another Ray lying between rays OP and OR . Prove that angle ROS = 1/2(angle QOS - angle POS).

Answer 1

Given- POQ is a line OR⊥PQ. OS is a line between OP and OR.
To prove that- ∠ROS-1/2 (∠QOS-∠POS)
proof- since, OR⊥PQ
therefore, ∠QOS = 90
since, PQ is a line OR stands on it
therefore, ∠POR+∠QOR = 180 (linear pair)
∠POR = 90
since, ∠POS + ∠SOR = 90 ...........(i)
∠QOR = 90
∠QOS - ∠ROS = 90 .................... (ii)
from eq (i) and (ii)
∠POS + ∠SOR = ∠QOS -∠POS
∠ROS + ∠ROS = ∠QOS - ∠POS
2∠ROS = ∠QOS-∠POS

∠ROS = 1/2(∠QOS - ∠POS)
Hence proved