In the diagram above, O is the centre of the circle $\angle UOT=70^\circ$ and $\angle RST=100^\circ$ Calculate $\angle RUO$
50o
20o
80o
25o
$\begin{align} & R\overset{\wedge }{\mathop{U}}\,T+R\overset{\wedge }{\mathop{S}}\,T={{180}^{\circ }}\text{ } \\ & \text{ }\!\!\{\!\!\text{ sum of opp}\text{.}\angle \text{s of cyclic quad=18}{{\text{0}}^{o}}\} \\ & R\overset{\wedge }{\mathop{U}}\,T+100={{180}^{\circ }} \\ & R\overset{\wedge }{\mathop{U}}\,T={{80}^{\circ }} \\ & UO=OT\text{ }\!\!\{\!\!\text{ radius of circle }\!\!\}\!\!\text{ } \\ & O\overset{\wedge }{\mathop{U}}\,T=U\overset{\wedge }{\mathop{T}}\,O\text{ }\!\!\{\!\!\text{ Base angles of issoscele }\Delta \text{ }\!\!\}\!\!\text{ } \\ & a+a+{{70}^{\circ }}={{180}^{\circ }}\text{ }\!\!\{\!\!\text{ sum of angles in a }\Delta \text{ }\!\!\}\!\!\text{ } \\ & a={{55}^{\circ }} \\ & R\overset{\wedge }{\mathop{U}}\,T+R\overset{\wedge }{\mathop{U}}\,O={{80}^{\circ }} \\ & R\overset{\wedge }{\mathop{U}}\,T={{80}^{\circ }}-{{55}^{\circ }}={{25}^{\circ }} \\\end{align}$