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 360°. 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

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

AC is the diagonal of the quadrilateral

AC splits the quadrilateral into two triangles ∆ABC and ∆ADC

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

So, ∠ABC + ∠ADC + ∠DCB + ∠CBA = 360°

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 180°

In triangle ADC

∠CAD + ∠DCA + ∠D = 180° —- (i)

In the triangle ABC

∠B + ∠BAC + ∠BCA = 180° —- (ii)

Add both the equations

∠CAD + ∠DCA + ∠D + ∠B + ∠BAC + ∠BCA = 180° + 180°

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

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

So, ∠D + ∠DAB + ∠BCD + ∠B = 360°

∠D + ∠A + ∠C + ∠B = 360°

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

### 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 85°, 100°, 60°?

**Solution:**

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

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

So, ∠A + ∠B + ∠C + ∠D = 360°

85° + 100° + 60° + x° = 360

245° + x° = 360°

x° = 360° – 245°

x° = 115°

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

**Example 2:**

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

**Solution:**

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

So, ∠C = ∠A, ∠B = ∠D

Sum of angles is 360°

∠A + ∠B + ∠C + ∠D = 360°

75° + ∠B + 75° + ∠D = 360°

150° + ∠B + ∠D = 360°

∠B + ∠D = 360° – 150°

∠B + ∠D = 210°

∠B + ∠B = 210°

2∠B = 210°

∠B = \(\frac { 210° }{ 2 } \)

∠B = 105°

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

**Example 3:**

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

**Solution:**

The given angles are ∠A = (3x + 2)°, ∠B = (x – 3)°, ∠C = (2x + 1)°, ∠D = 2(2x + 5)°

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

∠A + ∠B + ∠C + ∠D = 360°

(3x + 2)° + (x – 3)° + (2x + 1)° + 2(2x + 5)° = 360°

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 ∠A = (3x + 2)° = (3(35) + 2) = 105 + 2 = 107°

∠B = (x – 3)° = (35 – 3) = 32°

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

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

**Example 4:**

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

**Solution:**

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

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

∠A + ∠B + ∠C + ∠D = 360°

20.87° + 53.11° + 8.57° + x° = 360°

82.55° + x° = 360°

x = 360 – 82.55

x = 277.45°

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

### 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 360°.

**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 = 540°.