Question

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

It is also given that $\angle XYZ=64$ and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects $\angle ZYP$,
find $\angle XYQ$ and reflex $\angle QYP$.

Answer 1

From figure $1,$ $\angle ROS=\theta ,\angle QOS={90}^{\circ }+\theta ,\angle POS={90}^{\circ }-\theta$

$\angle QOS-\angle POS={90}^{\circ }+\theta -({90}^{\circ }-\theta )=2\theta =2\angle ROS$

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

From figure $2,\angle XYZ={64}^{\circ },\angle XYQ={64}^{\circ }+\alpha ,\angle QYP=\angle ZYQ=\alpha$

${64}^{\circ }+2\alpha ={180}^{\circ }⟹\alpha ={58}^{\circ }$

$\angle XYQ={64}^{\circ }+{58}^{\circ }={122}^{\circ }$

reflex $\angle QYP={360}^{\circ }-{58}^{\circ }={302}^{\circ }$