The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Show that the quadrilateral is a trapezium.

Let ABCD be the quadrilateral.

Given,

Angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6.

Let angles of quadrilateral be ∠A = 3x, ∠B = 4x, ∠C = 5x and ∠D = 6x.

We know that,

Sum of angles of a quadrilateral is 360°.

∴ 3x + 4x + 5x + 6x = 360°

⇒ 18x = 360°

⇒ x = $\dfrac{360°}{18}$ = 20°.

⇒ ∠A = 3x = 3 × 20° = 60°,

⇒ ∠B = 4x = 4 × 20° = 80°,

⇒ ∠C = 5x = 5 × 20° = 100° and

⇒ ∠D = 6x = 6 × 20° = 120°.

⇒ ∠A + ∠D = 60° + 120° = 180°,

⇒ ∠B + ∠C = 80° + 100° = 180°.

Since, ∠A and ∠D are supplementary and ∠B and ∠C are supplementary.

∴ AB || CD.

Since, one of the opposite sides of quadrilateral ABCD is parallel and all interior angles are unequal.

Hence, proved that quadrilateral is a trapezium.