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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).


Solution

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 

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