Question

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

Answer 1

Given: Let the four angles of quadrilateral ABCD be 3x, 4x, 5x and 6x respectively.

Step-wise calculation:

1. Sum of interior angles of a quadrilateral = 360°.

2. 3x + 4x + 5x + 6x = 360°

⇒ 18x = 360°

⇒ x = 20°

3. Hence, the angles are:

∠A = 3x = 60°

∠B = 4x = 80°

∠C = 5x = 100°

∠D = 6x = 120°

4. Check adjacent-angle sums along each leg:

∠A + ∠D

= 60° + 120°

= 180°

∠B + ∠C

= 80° + 100°

= 180°

Each pair of interior angles on the same side of a transversal between the two bases are supplementary.

Since each pair of interior angles on the same side are supplementary, the opposite sides AB and CD are parallel.

So, ABCD is a trapezium.