In figure, if PQ || ST, \(\angle{PQR}\) = \(110^\circ\) and \(\angle{RST}\) = \(130^\circ\) , find \(\angle{QRS}\) .

Construction : Draw a line parallel to ST through R.



As it is given that,
PQ || ST,\(\angle{PQR}\) = \(110^\circ\) and \(\angle{RST}\) = \(130^\circ\)
We can also say that, AB || PQ || ST

\(\because \) \(\angle{PQR}\) + \(\angle{QRA}\) = \(180^\circ\)
....(interior angles on the same side of transversal)
\(\therefore \) \(110^\circ\) + \(\angle{QRA}\) = \(180^\circ\)
\(\Rightarrow \) \(\angle{QRA}\) = \(70^\circ\)

\(\because \) \(\angle{ARS}\) = \(130^\circ\) ....(Alternate Interior angle)
As, \(\angle{RST}\) = \(130^\circ\)
Now, so as to find \(\angle{QRS}\), We have,
\(\angle{ARS}\) = \(\angle{ARQ}\) + \(\angle{QRS}\)
\(\Rightarrow \) \(130^\circ\) = \(70^\circ\) + \(\angle{QRS}\)
\(\Rightarrow \) \(\angle{QRS}\) = \(60^\circ\)