Question

In fig 3.31 , if $PQ||ST,\angle PQR={110}^{\circ }and\angle RST={130}^{\circ }$ , find $\angle QRS$
(Hint : Draw a line parallel to ST through point R )

Answer 1

Data : $PQ||ST$ and $\angle PQR={110}^{\circ }and\angle RST={130}^{\circ }$
To Prove : $\angle QRS=?$
Construction : Draw $ST||UV$ Through 'R'
Proof $PQ||St$ (Data )
$ST||UV$ (Construction )
$\therefore PQ||ST||UV$
$PQ||UV$
$\therefore \angle PQR+\angle URQ={180}^{\circ }$ (Sum of interior angles )
$110+\angle URQ={180}^{\circ }$
$\therefore URO=180-110$
$\therefore \angle URO={70}^{\circ }\left(i\right)$
Similarly , $ST||UV$
$\therefore \angle RST+\angle SRV=180$
$\therefore \angle SRV=180-130$
$\angle SRV={50}^{\circ }\left(ii\right)$
But , URV is a straight line .
$\angle QRS+50+70=180$

$\angle QRS=180-120$
$\therefore \angle QRS={60}^{\circ }$