if POQ is a line ray OR is perpendicular to line PQ OS is another layer lying between rays OP and OR prove that angle are OS is 1/2 (QOS - POS)
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To Prove: ROS = 1/2 (QOS – POS) .
POQ is a straight line
The sum of all angles made on it is 180°
=> ∠POS + ∠ROS + ∠ROQ = 180°
=> ∠POS + ∠ROS + 90° = 180° [given ∠ROQ = 90°]
=> ∠POS + ∠ROS = 90°
=> ∠ROS = 90° – ∠POS –eq(i)
∠ROS + ∠ROQ = ∠QOS [from figure]
=> ∠ROS + 90° = ∠QOS
=> ∠ROS = ∠QOS – 90° –eq(ii)
On Adding both the equations eq(i) + eq(ii)
=> ∠ROS + ∠ROS = 90° – ∠POS + ∠QOS – 90°
=> 2∠ROS =(∠QOS – ∠POS)
=> ∠ROS = (1/2) (∠QOS – ∠POS)
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