# Angle Sum Property of Quadrilateral – Theorem, Proof & Examples

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A quadrilateral is a closed shape and a polygon with four sides, four vertices, and four angles. It is formed by joining four non-collinear points. Irrespective of the shape and size of a quadrilateral, the sum of its four angles always remains the same. This is called the angle sum property of a quadrilateral.

Let’s understand what is angle sum property of quadrilateral with examples.

## Angle Sum Property of Quadrilateral

A quadrilateral is a 2D geometric figure with four sides, four angles, and four vertices. The angle sum property of quadrilateral states that the sum of all the four interior angles of a quadrilateral is $360^{\circ}$.

In the above figure, for a quadrilateral $\text{ABCD}$, $\angle \text{A} + \angle \text{B} + \angle \text{C} + \angle \text{D} = 360^{\circ}$.

### Proof of Angle Sum Property of Quadrilateral

To prove that the sum of four interior angles of a quadrilateral is $360^{\circ}$, let’s consider a quadrilateral $\text{ABCD}$, with diagonal $\text{AC}$.

The diagonal $\text{AC}$ divides the quadrilateral $\text{ABCD}$ into two triangles $\text{ABC}$ and $\text{ACD}$.

In $\triangle \text{ABC}$, $\angle \text{CAB} + \angle \text{ABC} + \angle \text{BCA} = 180^{\circ}$ (Sum of three interior angles of a triangle is $180^{\circ}$) ——————————— (1)

In $\triangle \text{ACD}$, $\angle \text{DAC} + \angle \text{ACD} + \angle \text{CDA} = 180^{\circ}$ (Sum of three interior angles of a triangle is $180^{\circ}$) ——————————— (2)

$\angle \text{CAB} + \angle \text{ABC} + \angle \text{BCA} + \angle \text{DAC} + \angle \text{ACD} + \angle \text{CDA} = 180^{\circ} + 180^{\circ}$

$=> (\angle \text{CAB} + \angle \text{DAC}) + \angle \text{ABC} + (\angle \text{BCA} + \angle \text{ACD}) + \angle \text{CDA} = 360^{\circ}$

$=> \angle \text{DAB} + \angle \text{ABC} + \angle \text{BCD} + \angle \text{CDA} = 360^{\circ}$

$=> \angle \text{A} + \angle \text{B} + \angle \text{C} + \angle \text{D} = 360^{\circ}$

### Examples on Angle Sum Property of Quadrilateral

Example 1: For which of the following angles, quadrilateral is not possible?

1. $100^{\circ}$, $60^{\circ}$, $120^{\circ}$, $80^{\circ}$
2. $100^{\circ}$, $80^{\circ}$, $60^{\circ}$, $70^{\circ}$
3. $110^{\circ}$, $70^{\circ}$, $80^{\circ}$, $110^{\circ}$
4. $110^{\circ}$, $50^{\circ}$, $120^{\circ}$, $80^{\circ}$
1. $100^{\circ}$, $60^{\circ}$, $120^{\circ}$, $80^{\circ}$

Sum of angles = $100^{\circ} + 60^{\circ} + 120^{\circ} + 80^{\circ} = 360^{\circ}$

Since the sum of four interior angles is $360^{\circ}$, therefore quadrilateral is possible.

1. $100^{\circ}$, $80^{\circ}$, $60^{\circ}$, $70^{\circ}$

Sum of angles = $100^{\circ} + 80^{\circ} + 60^{\circ} + 70^{\circ} = 310^{\circ}$

Since the sum of four interior angles is not $360^{\circ}$, therefore quadrilateral is not possible.

1. $110^{\circ}$, $70^{\circ}$, $80^{\circ}$, $110^{\circ}$

Sum of angles = $110^{\circ} + 70^{\circ} + 80^{\circ} + 110^{\circ} = 370^{\circ}$

Since the sum of four interior angles is not $360^{\circ}$, therefore quadrilateral is not possible.

1. $110^{\circ}$, $50^{\circ}$, $120^{\circ}$, $80^{\circ}$

Sum of angles = $110^{\circ} + 50^{\circ} + 120^{\circ} + 80^{\circ} = 360^{\circ}$

Since the sum of four interior angles is not $360^{\circ}$, therefore quadrilateral is not possible.

Example 2: Three angles of a quadrilateral are $100^{\circ}$, $80^{\circ}$, and $75^{\circ}$. Find the measure of the fourth angle.

Let the measure of the fourth angle of a quadrilateral be $x$ (in degrees).

Therefore $255 + x = 360$

$=> x = 360 – 255$

$=> x = 105$

Thus the measure of the fourth angle of the quadrilateral is $105^{\circ}$.

Example 3: The measure of four interior angles of a quadrilateral are in the ratio $6: 4 : 3: 2$. Find the sum of the largest two angles.

The measure of four interior angles of a quadrilateral are in the ratio $6 : 4 : 3 : 2$.

Therefore let the measure of four angles be $6x$, $4x$, $3x$, and $2x$ (in degrees)

According to angle sum property of quadrilateral $6x + 4x + 3x + 2x = 360$

$=> 15x = 360$

$=> x = \frac{360}{15} => x = 24$

Thus the measure of the four angles of quadrilateral is

$6 \times 24 = 144^{\circ}$

$4 \times 24 = 96^{\circ}$

$3 \times 24 = 72^{\circ}$

$2 \times 24 = 48^{\circ}$

Thus the sum of the two largest angles is $144 + 96 = 240^{\circ}$.

Example 4: Find the value of $x$ and $y$.

According to angle sum property of quadrilateral, $\angle \text{P} + \angle \text{Q} + \angle \text{R} + \angle \text{S} = 360$

$=> x + y + 20 + x + 100 + 80 + y = 360$

$=> 2x + 2y + 200 = 360$

$=> 2x + 2y = 360 – 200$

$=> 2x + 2y = 160$

$=> 2(x + y) = 160$

$=> x + y = \frac{160}{2}$

$=> x + y = 80$ ———————————————— (1)

Also $130 + 20 + x = 180$ (Sum of interior angle and exterior angle is $180^{\circ}$)

$x + 150 = 180$

$=> x = 180 – 150$

$=> x = 30$

Substituting $x = 30$ in (1)

$30 + y = 80$

$=> y = 80 – 30$

$=> y = 50$

Example 5: What is the value of $\angle \text{POQ}$ if $\text{OP}$ and $\text{OQ}$ are angle bisectors?

In the above quadrilateral $\text{PQRS}$, $\angle \text{P} + \angle \text{Q} + \angle \text{R} + \angle \text{S} = 360^{\circ}$

$=> \angle \text{P} + \angle \text{Q} + 60 + 110 = 360$

$=> \angle \text{P} + \angle \text{Q} + 170 = 360$

$=> \angle \text{P} + \angle \text{Q} = 360 – 170$

$=> \angle \text{P} + \angle \text{Q} = 190$

Therefore $\angle \text{OPQ} + \angle \text{PQO} = \frac{1}{2}\angle \text{P} + \frac{1}{2}\angle \text{Q}$ ($\text{OP}$ and $\text{OQ}$ are the angle bisectors of $\angle \text{P}$, and $\angle \text{Q}$.

$=> \angle \text{OPQ} + \angle \text{PQO} = \frac{1}{2} \times 190$

$=> \angle \text{OPQ} + \angle \text{PQO} = 95$

In $\triangle \text{OPQ}$, $\angle \text{OPQ} + \angle \text{PQO} + \angle \text{POQ} = 180^{\circ}$ (Angle sum property of triangle)

$=> 95 + \angle \text{POQ} = 180$

$=> \angle \text{POQ} = 180 – 95$

$=> \angle \text{POQ} = 85^{\circ}$

## Practice Problems

1. If the sum of three interior angles of a quadrilateral is $240^{\circ}$, find the fourth angle.
2. If the angles of a quadrilateral are in the ratio $6:3:4:5$, determine the value of the four angles.
3. In a quadrilateral, if the sum of two angles is $240^{\circ}$, find the measure of the other two equal angles if they are in ratio $1: 3$.
4. If three angles of a quadrilateral are equal and the measure of the fourth angle is $30^{\circ}$, find the measure of each of the equal angles.
5. If one angle of a quadrilateral is double of another angle and the measure of the other two angles are $60^{\circ}$, $80^{\circ}$. Find the measurement of the unknown angles.

## FAQs

### Explain the angle sum property of quadrilateral.

The angle sum property of quadrilateral states that the sum of all the four interior angles of quadrilateral is $360^{\circ}$.

### What is the measure of the sum of the interior angles of a quadrilateral?

The sum of all the four interior angles of a quadrilateral is $360^{\circ}$.

### How to prove the angle sum property of a quadrilateral?

To prove the angle sum property of a quadrilateral, we construct a diagonal joining any two opposite vertices of it. The diagonal divides the quadrilateral into two triangles. Then using the angle sum property of a triangle, we prove the angle sum property of quadrilateral.

Note: The sum of three interior angles of a triangle is $180^{\circ}$.

### What is the measure of the sum of the exterior angles of a quadrilateral?

The measure of the sum of all the four exterior angles of a quadrilateral is $360^{\circ}$.

Note: The sum of all the exterior angles of any polygon is $360^{\circ}$.

## Conclusion

A quadrilateral is a 2D geometric figure with four sides, four angles, and four vertices. The angle sum property of quadrilateral states that the sum of all the four interior angles of a quadrilateral is $360^{\circ}$. The angle sum property of quadrilateral helps in solving problems based on quadrilaterals.