Question

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 ______.

पर्याय

• rhombus

• parallelogram

• trapezium

• kite

Answer 1

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 trapezium.

Explanation:

Given in the question, ratio of angles of quadrilateral ABCD is 3 : 7 : 6 : 4.

Let the angles of quadrilateral ABCD be 3x, 7x, 6x and 4x respectively.

So, 3x + 7x + 6x + 4x = 360°  ...[Sum of the all angles of a quadrilateral is 360°]

20x = 360°

x = `(360^circ)/20`

x = 18°

So, angles of the quadrilateral are:

∠A = 3 × 18° = 54°

∠B = 7 × 18° = 126°

∠C = 6 × 18° = 108°

∠D = 4 × 18° = 72°

See the figure, ∠BCE = 180° – ∠BCD  ...[Linear pair axiom]

∠BCE = 180° – 108° = 72°

∠BCE = ∠ADC = 72°

Now, BC || AD   ...[The corresponding angles are equal]

The sum of co interior angles is:

∠A + ∠B = 126° + 54° = 180°

And ∠C + ∠D = 108° + 72° = 180°

Hence, ABCD is a trapezium.