# Sum of Angles of a Quadrilateral | Quadrilateral Angles Sum Property – Theorem, Proof, Examples

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.

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

AC is the diagonal of the quadrilateral

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°

âˆ CAD + âˆ DCA + âˆ D = 180Â° —- (i)

In the triangle ABC

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

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°

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.

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°.

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