Question

Ratio of consecutive angles of a quadrilateral is 1 : 2 : 3 : 4. Find the measure of its each angle. Write, with reason, what type of a quadrilateral it is.

Answer 1

Suppose PQRS is a quadrilateral.

Let m∠P : m∠Q : m∠R : m∠S = 1 : 2 : 3 : 4

So, m∠P = k, m∠Q = 2k, m∠R = 3k and m∠S = 4k, where k is some constant

Now,

m∠P + m∠Q  + m∠R + m∠S = 360°

∴ k + 2k + 3k + 4k = 360°

⇒ 10k = 360°

⇒ k = 36°

∴ m∠P = 36°

m∠Q = 2k = 2 × 36° = 72°

m∠R = 3k = 3 × 36° = 108°

m∠S = 4k = 4 × 36° = 144°

Now,  m∠P + m∠S = 36° + 144° = 180°

We know if two lines are intersected by a transversal such that the sum of interior angles on the same transversal is supplementary, then the two lines are parallel.

∴ Side PQ || Side SR

Also, m∠P + m∠Q = 36° + 72° = 108° ≠ 180°

So, side PS is not parallel to side QR.

In quadrilateral PQRS, only one pair of opposite sides is parallel. Therefore, quadrilateral PQRS is a trapezium.