In Fig. 8.48, 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.)

OR perpendicular to PQ

Therefore,

∠POR = 90^o

∠POS + ∠SOR = 90^o [Therefore, ∠POR = ∠POS + ∠SOR]

∠ROS = 90^o - ∠POS (i)

∠QOR = 90^o (Therefore, OR perpendicular to PQ)

∠QOS - ∠ROS = 90^o

∠ROS = ∠QOS – 90^o(ii)

By adding (i) and (ii), we get

2∠ROS = ∠QOS - ∠POS

∠ROS = (∠QOS - ∠POS)