# Angles of a Quadrilateral are in Ratio

The four angles of a quadrilateral are in ratio then how to find the measure of each angle of the quadrilateral.  According to the angle sum property of quadrilateral, we know that the sum of the angles of a quadrilateral is 360°.

Solved examples of angles of a quadrilateral are in ratio:

1. In a quadrilateral ABCD, the angles A, B, C, D are in the ratio 3 : 5 : 7 : 9. Find the measure of each angle of the quadrilateral.

Solution:

Let the common ratio be x.

Then the four angles of the quadrilateral are 3x, 5x, 7x, 9x.

According to the angle sum property of quadrilateral,

3x + 5x + 7x + 9x = 360

⇒ 24x = 360

⇒ x = 360/24

⇒ x = 15°

Therefore, measure of angle A 3x = 3 × 15 = 45°

Measure of angle B = 5x = 5 × 15 = 75°

Measure of angle C = 7x = 7 × 15 = 105°

Measure of angle D = 9x = 9 × 15 = 135°

Therefore, the four angles of the quadrilateral are 45°, 75°, 105° and 135°.

2. The four angles of a quadrilateral are in the ratio 2 : 3 : 5 : 8. Find the angles.

Solution:

Let the measures of angles of the given quadrilateral be (2x)°, (3x)°, (5x)° and (8x)°.

We know that the sum of the angles of a quadrilateral is 360°.

Therefore, 2x + 3x + 5x + 8x = 360

⇒ 18x = 360

⇒ x = 20.

So, the measures of angles of the given quadrilateral are

(2 × 20)°, (3 × 20)°, (5 × 20)° and (8 × 20)°

i.e., 40°, 60°, 100° and 160°.

3. The angles of a quadrilateral are in ratio 1 : 2 : 3 : 4. Find the measure of each of the four angles.

Solution:

Let the common ratio be x.

Then the measure of four angles is 1x, 2x, 3x, 4x

We know that the sum of the angles of quadrilateral is 360°.

Therefore, x + 2x + 3x + 4x = 360°

⇒ 10x = 360°

⇒ x = 360/10

⇒ x = 36

Therefore, 1x = 1 × 36 = 36°

2x = 2 × 36 = 72°

3x = 3 × 36 = 108°

4x = 4 × 36 = 144°

Hence, the measure of the four angles is 36°, 72°, 108°, and 144°

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