Question

In the fig 2.59, if PQ $\perp$ PS, PQ || SR, $\angle$SQR = 30$°$, $\angle$QRT = 65$°$ then find the values of x, y, and z.

Answer 1

$$\begin{gathered} \mathrm{Given}:\angle \mathrm{SQR}=30°\mathrm{and}\angle \mathrm{QRT}=65° \\ \angle \mathrm{QRT}+\angle \mathrm{QRS}=180°(\mathrm{Linear}\mathrm{pair}) \\ \Rightarrow 65°+\angle \mathrm{QRS}=180° \\ \Rightarrow \angle \mathrm{QR}S=115° \\ \mathrm{In}△\mathrm{QRS},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{SQR}+\angle \mathrm{QRS}+\angle \mathrm{QSR}=180° \\ \Rightarrow 30°+115°+\angle \mathrm{QSR}=180° \\ \Rightarrow \angle \mathrm{QSR}=35°...\left(\mathrm{i}\right) \\ \mathrm{As}x=\angle \mathrm{QSR}=35°(\because \mathrm{PQ}\parallel \mathrm{SR}) \\ \mathrm{In}△\mathrm{PQS},\mathrm{we}\mathrm{have}: \\ x+y+90°=180° \\ \Rightarrow 35°+y+90°=180°\mathrm{from}\left(\mathrm{i}\right) \\ \Rightarrow y=55°...\left(\mathrm{ii}\right) \\ \mathrm{As}\mathrm{PSV}\mathrm{is}\mathrm{a}\mathrm{straight}\mathrm{line},\mathrm{we}\mathrm{have}: \\ y+\angle \mathrm{QSR}+z=180° \\ \Rightarrow 55°+35°+z=180°\left[\mathrm{From}\right(\mathrm{i})\mathrm{and}(\mathrm{ii}\left)\right] \\ \Rightarrow z=90° \end{gathered}$$