A quadrilateral can be defined as a closed geometric, two-dimensional shape having 4 straight sides. It has 4 vertices and angles. The types of quadrilaterals are parallelograms, squares, rhombus, and rectangle. The sum of all interior angles of a quadrilateral is equal to 360Â°. The angle is formed when two line segments meet at a common point. The angle can be measured in degrees or radians. The angles of a quadrilateral are the angles formed inside the closed shape.

## Sum of Angles of a Quadrilateral Theorem & Proof

The sum of interior angles of a quadrilateral is 360 degrees.

In the quadrilateral ABCD

âˆ ABC, âˆ ADC, âˆ DCB, âˆ CBA are the interior angles

AC is the diagonal of the quadrilateral

AC splits the quadrilateral into two triangles âˆ†ABC and âˆ†ADC

We know that sum of angles of a quadrilateral is 360Â°

So, âˆ ABC + âˆ ADC + âˆ DCB + âˆ CBA = 360Â°

Let’s prove that sum of all interior angles of a quadrilateral is 360 degrees.

We know that the sum of angles in a triangle is 180Â°

In triangle ADC

âˆ CAD + âˆ DCA + âˆ D = 180Â° —- (i)

In the triangle ABC

âˆ B + âˆ BAC + âˆ BCA = 180Â° —- (ii)

Add both the equations

âˆ CAD + âˆ DCA + âˆ D + âˆ B + âˆ BAC + âˆ BCA = 180Â° + 180Â°

âˆ D + (âˆ CAD + âˆ BAC) + (âˆ BCA + âˆ DCA) + âˆ B = 360Â°

We can see that âˆ CAD + âˆ BAC = âˆ DAB, âˆ BCA + âˆ DCA = âˆ BCD

So, âˆ D + âˆ DAB + âˆ BCD + âˆ B = 360Â°

âˆ D + âˆ A + âˆ C + âˆ B = 360Â°

Therefore, the sum of angles of a quadrilateral is 360Â°

### Quadrilateral Angles Sum Propoerty

Each quadrilateral has 4 angles. The sum of its interior angles is always 360 degrees. So, we can find the angles of the quadrilateral if we know the remaining 3 angles or 2 angles or 1 angle and 4 sides. For a square or rectangle, the value of all angles is 90 degrees.

**Also, Read**

- What is a Quadrilateral?
- Construct Different Types of Quadrilaterals
- Different Types of Quadrilaterals

### Examples on Quadrilateral Angles

**Example 1:**

Find the fourth angle of the quadrilateral if three angles are 85Â°, 100Â°, 60Â°?

**Solution:**

The given three angles of a quadrilateral are 85Â°, 100Â°, 60Â°

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

So, âˆ A + âˆ B + âˆ C + âˆ D = 360Â°

85Â° + 100Â° + 60Â° + xÂ° = 360

245Â° + xÂ° = 360Â°

xÂ° = 360Â° – 245Â°

xÂ° = 115Â°

Therefore, the fourth angle of the quadrilateral is 115Â°.

**Example 2:**

Find the measure of the missing angles in a parallelogram if âˆ A = 75Â°?

**Solution:**

We know that the opposite angles of a parallelogram are equal.

So, âˆ C = âˆ A, âˆ B = âˆ D

Sum of angles is 360Â°

âˆ A + âˆ B + âˆ C + âˆ D = 360Â°

75Â° + âˆ B + 75Â° + âˆ D = 360Â°

150Â° + âˆ B + âˆ D = 360Â°

âˆ B + âˆ D = 360Â° – 150Â°

âˆ B + âˆ D = 210Â°

âˆ B + âˆ B = 210Â°

2âˆ B = 210Â°

âˆ B = \(\frac { 210Â° }{ 2 } \)

âˆ B = 105Â°

So, other angles of a parallelogram are 105Â°, 75Â°, 105Â°.

**Example 3:**

The angle of a quadrilateral are (3x + 2)Â°, (x â€“ 3)Â°, (2x + 1)Â°, 2(2x + 5)Â° respectively. Find the value of x and the measure of each angle?

**Solution:**

The given angles are âˆ A = (3x + 2)Â°, âˆ B = (x â€“ 3)Â°, âˆ C = (2x + 1)Â°, âˆ D = 2(2x + 5)Â°

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

âˆ A + âˆ B + âˆ C + âˆ D = 360Â°

(3x + 2)Â° + (x â€“ 3)Â° + (2x + 1)Â° + 2(2x + 5)Â° = 360Â°

3x + 2 + x – 3 + 2x + 1 + 4x + 10 = 360

10x + 10 = 360

10x = 360 – 10

10x = 350

x = \(\frac { 350 }{ 10 } \)

x = 35

The measurement of each angle of a quadrilateral is âˆ A = (3x + 2)Â° = (3(35) + 2) = 105 + 2 = 107Â°

âˆ B = (x â€“ 3)Â° = (35 – 3) = 32Â°

âˆ C = (2x + 1)Â° = (2(35) + 1) = 70 + 1 = 71Â°

âˆ D = 2(2x + 5)Â° = 2(2(35) + 5) = 2(70 + 5) = 2(75) = 150Â°

**Example 4:**

The three angles of a closed 4 sided geometric figure are 20.87Â°, 53.11Â°, 8.57Â°. Find the fourth angle?

**Solution:**

The given angles are âˆ A = 20.87Â°, âˆ B = 53.11Â°, âˆ C = 8.57Â°

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

âˆ A + âˆ B + âˆ C + âˆ D = 360Â°

20.87Â° + 53.11Â° + 8.57Â° + xÂ° = 360Â°

82.55Â° + xÂ° = 360Â°

x = 360 – 82.55

x = 277.45Â°

Therefore, the fourth angle of the closed 3 sided geometric figure is 277.45Â°.

### FAQs on Sum of Angles of a Quadrilateral

**1. What is the sum of the internal angles of a quadrilateral?**

The sum of angles of a quadrilateral is 360 degrees.

**2. What are the properties of a quadrilateral?**

The three different properties of a quadrilateral are it has four sides, four vertices, four angles. And it is a closed 2-dimensional geometric figure. The sum of within angles is 360 degrees.

**3. How do you prove the angle sum property of a quadrilateral?**

To prove the sum property of a quadrilateral, draw a diagonal to divide it into two triangles. The sum of all interior angles of a triangle is 180 degrees. thus, the sum of angles of a quadrilateral becomes 360Â°.

**4. What is the sum of all interior angles of a pentagon?**

Draw one diagonal that should divide the pentagon into one triangle, one quadrilateral. The sum of angles of a triangle is 180 degrees, the sum of angles of a quadrilateral is 360 degrees. So, the sum of all interior angles of a pentagon is 180 + 360 = 540Â°.