Question

In quadrilateral PQRS, ∠P : ∠Q : ∠R : ∠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

∵ ∠P : ∠Q : ∠R : ∠S = 3: 4: 6: 7 

Let ∠P = 3x

∠Q = 4x

∠R = 6x

∠S = 7x

∴ ∠P + ∠Q + ∠R + ∠S = 360°

3x + 4x + 6x + 7x = 360°

20x = 360°

x = 18°

∴ ∠P = 3x = 3 × 18 = 54°

∠Q = 4x = 4 × 18 = 72°

∠R = 6x = 6 × 18 = 108°

∠S = 7x = 7 × 18 = 126°

∠Q + ∠R = 72° + 108° = 180°

or ∠P + ∠S = 54° + 126° = 180°

Hence PQ || SR

As ∠P + ∠Q = 72° + 54° = 126° 

Which is ≠ 180°

∴ PS and QR are not parallel.

PQRS is a Trapezium as its one pair of opposite side is parallel.