In the diagram P, Q, R, S are points on the circle $R\overset{\wedge }{\mathop{Q}}\,S$ = 30o, $P\overset{\wedge }{\mathop{R}}\,S$ = 50o and $P\overset{\wedge }{\mathop{S}}\,Q$ = 20o. What is the value of xo + yo ?
260o
130o
100o
80o
$\begin{align} & \angle SPR=\angle SQR=30\text{ }\!\!\{\!\!\text{ }\angle s\text{ in the same segment }\!\!\}\!\!\text{ } \\ & \text{Consider }\vartriangle PSR \\ & \angle SPR+\angle PSR+\angle SRP={{180}^{\circ }} \\ & {{30}^{\circ }}+{{(20+x)}^{\circ }}+{{50}^{\circ }}={{180}^{\circ }} \\ & x={{80}^{\circ }} \\ & \text{Consider }\vartriangle SRZ \\ & \angle ZSR+\angle SRZ+\angle SZR={{180}^{\circ }} \\ & \angle SZY={{50}^{\circ }} \\ & y=\angle PZY=\angle SZY={{50}^{\circ }}\text{ vertically opposites }\angle s\} \\ & x+y={{50}^{\circ }}+{{80}^{\circ }}={{130}^{\circ }} \\\end{align}$