Question

In the figure line RS, VP and TQ intersect in point 0, $\angle VOS={100}^{\circ },\angle ROT={122}^{\circ }$. Find $\angle ROQ,\angle QOP$ and $\angle POS$.

Answer 1

$Giventhat\angle ROT={122}^{o}and\angle VOS={100}^{o}SinceVP,RSandCTintersectingatpointOhencewecanwrite\angle ROV=\angle POS\left(byVOA\right)..........................\left(1\right)\angle ROQ=\angle SOT\left(byVOA\right)----------\left(2\right)Soweget\angle ROT=\angle ROV+\angle VOT=\angle POS+\angle VOT\left(from1above\right)Since\angle ROT={122}^o,wecanwrite\angle POS+\angle VOT={122}^{o}Now,wehave\angle POS+\angle SOT+\angle VOT={180}^{o}Substitutingthevalueof\angle POS+\angle VOT,weget{122}^o+\angle SOT={180}^{o}Therefore\angle SOT={180}^o-{122}^o={58}^{o}From2above\angle ROQ={58}^{o}Given,\angle VOS={100}^{o}So,\angle POS=180-\angle VOS=180-100=80=\angle ROV\left(byVOA\right)GIven\angle ROT=\angle ROV+\angle VOT={122}^{o}As\angle ROV=80,weget\angle VOT=122-80=42=\angle QOP\left(byVOA\right)Therefore,\angle ROQ={58}^o,\angle POS={80}^{o}and\angle QOP={42}^o$