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.

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Solution

## 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 = 360º ∴ k + 2k + 3k + 4k = 360º ⇒ 10k = 360º ⇒ k = 36º ∴ m∠P = 36º m∠Q = 2k = 2 × 36º = 72º m∠R = 3k = 3 × 36º = 108º m∠S = 4k = 4 × 36º = 144º Now, m∠P + m∠S = 36º + 144º = 180º We know if two lines are intersected by a transversal such that the sum of interior angles on the same of the transversal are supplementary, then the two lines are parallel. ∴ Side PQ || Side SR Also, m∠P + m∠Q = 36º + 72º = 108º ≠ 180º 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.

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