Question

Question 4
In the given figure, if $PQ||ST,\angle PQR={110}^{\circ }$ and $\angle RST={130}^{\circ }$ find the $\angle QRS.$
[Hint: Draw a line parallel to ST through point R.]

Answer 1

Let us draw a line XY parallel to ST and passing through point R.
$\angle PQR+\angle QRX={180}^{\circ }$ (Co-interior angles on the same side of transversal QR)
Then ${110}^{\circ }+\angle QRX={180}^{\circ }$
$\therefore \angle QRX=180-110={70}^{\circ }$
Also,
$\angle RST+\angle SRY={180}^{\circ }$ (Co-interior angles on the same side of transversal SR)
${130}^{\circ }+\angle SRY={180}^{\circ }$
$\angle SRY=180-130={50}^{\circ }$
XY is a straight line, RQ and RS stand on it.
$\therefore \angle QRX+\angle QRS+\angle SRY={180}^{\circ }$
${70}^{\circ }+\angle QRS+{50}^{\circ }={180}^{\circ }$
$\therefore \angle QRS={180}^{\circ }-{120}^{\circ }={60}^{\circ }$