In the given figure, if QP || ST, then find the measure of \(\angle PQR\).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 72°
The angles at Q and at S sit on two different parallel lines, so a third parallel line through R is drawn. Let RU be the line through R parallel to QP and ST, drawn towards the same side as T. RS then splits \(\angle QRU\) into \(\angle QRS=36^{\circ}\) and \(\angle SRU\).
The angles at R:
QP \(\parallel\) RU with transversal QR: \(\angle QRU=\angle PQR=x^{\circ}\) (alternate angles)
ST \(\parallel\) RU with transversal SR: \(\angle SRU=180^{\circ}-2x^{\circ}\) (co-interior angles with \(\angle TSR=2x^{\circ}\))
\(\angle QRU=\angle QRS+\angle SRU\), since RS lies between RQ and RU
Finding x:
\(x=36+(180-2x)\)
\(3x=216\)
\(x=72\), so \(\angle PQR=72^{\circ}\)