Question

In the following figure, if PQ || ST, ∠PQR = 110° and ∠RST = 130° find ∠QRS.

Answer 1

Extend line PQ to B intersecting RS at A.

AT and AB are parallel lines therefore
$$\begin{gathered} \angle SAB+\angle TSA={180}^{\circ } \\ \Rightarrow \angle SAB+{130}^{\circ }={180}^{\circ }\left(\because ={130}^{\circ }\right) \\ \Rightarrow \angle SAB={180}^{\circ }-{130}^{\circ }={50}^{\circ } \end{gathered}$$
$\therefore \angle QAR=\angle SAB={50}^{\circ }\left(\mathrm{Vertically}\mathrm{opposite}\mathrm{angles}\right)$
Now, in $∆QRA$
$$\begin{gathered} \angle QRA+\angle QAR+\angle AQR={180}^{\circ } \\ \Rightarrow \angle QRA+{50}^{\circ }+{180}^{\circ }-{110}^{\circ }={180}^{\circ } \\ \Rightarrow \angle QRA={60}^{\circ } \end{gathered}$$
Hence, $\angle QRS={60}^{\circ }$.