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

Since it is given that, OR is perpendicular to PQ, we have,\(\angle{PQR}\) = \(\angle{ROQ}\) = \(90^\circ\)
Also, we can say that,
\(\angle{POS}\) + \(\angle{ROS}\) = \(90^\circ\)
\(\Rightarrow \) \(\angle{ROS}\) = \(90^\circ\) - \(\angle{POS}\) ...(i)

Now, by adding, \(\angle{ROS}\) on both the side, we get,

2\(\angle{ROS}\) = \(90^\circ\) - \(\angle{POS}\) + \(\angle{ROS}\)
\(\Rightarrow \) 2\(\angle{ROS}\) = (\(90^\circ\) + \(\angle{ROS}\)) - \(\angle{POS}\)
\(\Rightarrow \) 2\(\angle{ROS}\) = \(\angle{QOS}\) - \(\angle{POS}\)
\(\Rightarrow \) \(\angle{ROS}\) = \(\frac{1}{2} \) (\(\angle{QOS}\) - \(\angle{POS}\)).
Hence, proved.