Question

In the figure 1.78 line RS, VP and TQ intersect in point O, $\angle$VOS = 100$°$, $\angle$ROT = 122$°$. Find $\angle$ROQ, $\angle$QOP and $\angle$POS.

Answer 1

Given: $\angle \mathrm{VOS}=100°,\angle \mathrm{ROT}=122°\mathrm{and}\angle \mathrm{VOT}=x$

$$\begin{gathered} \mathrm{Here},\angle \mathrm{VOR}+x+\angle \mathrm{TOS}=180°(\mathrm{Angles}\mathrm{on}\mathrm{a}\mathrm{straight}\mathrm{line}) \\ \Rightarrow 122°+\angle \mathrm{TOS}=180°\left(\mathrm{Given}\right) \\ \Rightarrow \angle \mathrm{TOS}=58°....\left(\mathrm{i}\right) \\ \mathrm{Again},\angle \mathrm{VOR}+x+\angle \mathrm{TOS}=180° \\ \Rightarrow \angle \mathrm{VOR}+100°=180° \\ \Rightarrow \angle \mathrm{VOR}=80°.....\left(\mathrm{ii}\right) \\ \mathrm{Similarly},\angle \mathrm{VOR}+x+\angle \mathrm{TOS}=180° \\ \Rightarrow 80°+x+58°=180° \\ \Rightarrow x=42°....\left(\mathrm{iii}\right) \\ \mathrm{By}\mathrm{vertically}\mathrm{opposite}\mathrm{angle}\mathrm{theorem},\mathrm{we}\mathrm{have}, \\ \angle \mathrm{ROQ}=\angle \mathrm{TOS}=58° \\ \angle \mathrm{QOP}=x=42° \\ \angle \mathrm{POS}=\angle \mathrm{VOR}=80° \end{gathered}$$