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Adding equation (iii) and (iv), we get

∠2 + ∠4 + ∠B + ∠1 + ∠3 + ∠D = 360° Remember

⇒ (∠1 + ∠2) + ∠B + (∠3 + ∠4) + ∠D = 360° Sum of all the exterior angles of a
quadrilateral is also 360°.
⇒ ∠A + ∠B + ∠C + ∠D = 360° [Using (i) and (ii)]

Hence, the sum of all interior angles of a quadrilateral is 360°.

Example 11: The three angles of a quadrilateral are 60°, 70° and 90°. Find the fourth angle.
Solution: Let the measure of the fourth angle be x.

According to the angle sum property of a quadrilateral, Quick Check
the sum of all angles of a quadrilateral = 360°
Find the value of unknown angle in
60° + 70° + 90° + x = 360° the following quadrilateral:

220° + x = 360° 1. 50° 2. 75°
x x
130°
x = 360° – 220° = 140° 120°
55°
Thus, the fourth angle of the quadrilateral is 140°.

Example 12: If the four angles of a quadrilateral are in the ratio of 9 : 8 : 4 : 15, find the measures
of each angle.
Solution: The ratio of the angles of a quadrilateral is 9 : 8 : 4 : 15.

Let the measures of angles be 9x, 8x, 4x, and 15x.

According to the angle sum property of a quadrilateral, the sum of all angles of a quadrilateral = 360°

9x + 8x + 4x + 15x = 360° ⇒ 36x = 360°
360°
x = ⇒ x = 10°
36
Therefore, 9x = 9 × 10° = 90°, 8x = 8 × 10° = 80°, 4x = 4 × 10° = 40° and 15x = 15 × 10° = 150°.

Thus, the angles of the quadrilateral are 90˚, 80˚, 40˚ and 150˚.

Example 13: If the measures of two angles of a quadrilateral are 55° and 75°, and the other two
angles are equal, find the measure of each of the equal angles.

Solution: Let the measure of each of the equal angle be x.
According to the angle sum property of a quadrilateral,

55˚ + 75˚ + x + x = 360˚

130˚ + 2x = 360˚ Think and Answer
2x = 230˚ Is it possible to have a quadrilateral

230° whose angles are of measures 105°,
x = 160°, 60° and 45°?
2
x = 115˚
Thus, the measure of equal angles is 115˚ each.

65 Quadrilaterals