5. In Fig. 6.17, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that    \angle \textup{ROS} = \frac{1}{2}(\angle \textup{QOS} - \angle \textup{POS})

                

Answers (1)


Given that,
POQ is a line, OR \perp PQ and \angle ROQ is a right angle.
Now, \angle POS + \angleROS +  \angleROQ = 180^0  [since POQ is a straight line] 
\\\Rightarrow \angle POS + \angle ROS = 90^0\\ \Rightarrow \angle ROS = 90^0-\angle POS.............(i)
and, \angle ROS + \angle ROQ = \angle QOS
       \angle ROS = \angle QOS -90^0..............(ii)

Add the eq (i ) and eq (ii),  we get

\angle \textup{ROS} = \frac{1}{2}(\angle \textup{QOS} - \angle \textup{POS})

hence proved

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