Angle Sum Property of a Quadrilateral

The sum of the four angles of a quadrilateral is 360°

If we draw a diagonal in the quadrilateral, it divides it into two triangles.  

And we know the angle sum property of a triangle i.e. the sum of all the three angles of a triangle is 180°.

The sum of angles of \mathrm{\triangle ADC = 180^o} .

The sum of angles of \mathrm{\triangle ABC=180^o} .

By adding both we get \mathrm{\angle A + \angle B + \angle C + \angle D = 360^o}

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

Example

Find \angle A and \angle D, if BC || AD and \angle B = 52° and \angle C = 60° in the quadrilateral ABCD.

Solution:

Given BC ∥ AD, so ∠A and ∠B are consecutive interior angles.

So \angle A + \angle B = 180° (Sum of consecutive interior angles is 180°).

\angle B = 52°

\angle A = 180°- 52° = 128°

\angle A + \angle B + \angle C + \angle D = 360° (Sum of the four angles of a quadrilateral is 360°).

\angle C = 60°

128° + 52° + 60° + \angle D = 360°

\angle D = 120°

\therefore\angle A = 128° and \angle D = 120°.

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