Question

In quadrilateral PQRS, $\angle P:\angle Q:\angle R:\angle S=3:4:6:7$.

Calculate each angle of the quadrilateral and then prove that PQ and SR are parallel to each other.

(i) Is PS also parallel to QR ?

(ii) Assign a special name to quadrilateral PQRS.

Answer 1

$\because \angle P:\angle Q:\angle R:\angle S=3:4:6:7Let\angle P=3x\angle Q=4x\angle R=6x\&\angle S=7x\therefore \angle P+\angle Q+\angle R+\angle S={360}^{o}3x+4x+6x+7x={360}^{o}20x={360}^{o}x={18}^o\therefore \angle P=3x=3\times 18={54}^o\angle Q=4x=4\times 18={72}^{o}R=6x=6\times 18={108}^o\angle S=7x=7\times 18={126}^o\angle Q+\angle R={72}^o+{108}^o={180}^{o}or\angle P+\angle S={54}^o+{126}^o={180}^o$

Hence PQ || SR

Which is $\ne {180}^o.\therefore \text{PS and QR are not parallel}\left(ii\right)\text{PQRS is a Trapezium as its one pair of opposite side is parallel}$.