If angles A, B, C and D of the quadrilateral ABCD, taken in order, are in the ratio 3:7:6:4, then ABCD is a

Given: Ratio of angles is 3: 7: 6: 4

Let the angles be 3x, 7x, 6x and 4x.

Sum of all angles of a quadrilateral is 360°

⇒ 3x + 7x + 6x + 4x = 360°

⇒ 20x = 360°

⇒ x = 18°

Now,

Angles of quadrilateral are

∠A = 3x = 3 × 18° = 54°

∠B = 7x = 7 × 18° = 126°

∠C = 6x = 6 × 18° = 108°

∠D = 4x = 4 × 18° = 72°

Extend DC to E

Now,

∠BCE + ∠BCD = 180° [∵ sum of angles on a straight line = 180°]

⇒ ∠BCE + ∠BCD = 180°

⇒ ∠BCE + 108° = 180°

⇒ ∠BCE = 180° – 108°

⇒ ∠BCE = 72°

∵ ∠BCE = ∠ADC = 72°

∴ BC ∥ AD

Hence, it is a trapezium.