# Angles in a Quadrilateral – GCSE Math

In this chapter we will about about the angles of quadrilateral, also the method to find the missing angle if measure of other angles are given.

## What are angles in a quadrilateral ?

It’s the interior angle formed between the two adjacent sides. In Quadrilateral, there are 4 interior angles present.

Consider the above quadrilateral ABCD in which ∠1, ∠2, ∠3 and ∠4 are the interior angles of quadrilateral.

∠1 is formed between side DA & AB
∠2 is formed between side CB & AB
∠3 is formed between side BC & DC
∠4 is formed between side AD & CD

### Sum of angles of Quadrilateral

The sum of all four angles of quadrilateral measures 360 degree.

Please remember this as a rule as it would help you solve lots of geometry related problems.

Let’s again consider the same quadrilateral ABCD.

Using the angle sum property rule of quadrilateral we can write;

∠1+ ∠2 + ∠3 + ∠4 = 360 degree

### Proof of angle sum property of quadrilateral

Take the above quadrilateral ABCD and join one of its diagonal AC.

Now when we join the diagonal AC, we get two triangles ABC and ADC.

We know that sum of all angles of triangle measures 180 degree.

Sum ( all angles of triangle ABC ) = 180 degree

Sum ( all angles of triangle CDA ) = 180 degree

Now if we add angles of both the triangle we will get to total of 360 degrees.

Sum ( triangle ABC + triangle CDA ) = 180 + 180

Sum ( quadrilateral ABCD) = 360

### Angle property of different quadrilateral

#### Angle Property of Square

In square, all interior angles are right angles.

#### Angle property of rectangle

In rectangle, all interior angles are right angle (i.e. 90 degree)

#### Angle property of Parallelogram

In parallelogram, opposite angles are equal to each other.
∠A = ∠C
∠D = ∠B

Also, sum of supplementary angles measure 180 degree
∠A + ∠B = 180 degree
∠B + ∠C = 180 degree
∠C + ∠D = 180 degree
∠D + ∠A = 180 degree

#### Angle property of Rhombus

The angle property of rhombus is same as that of parallelogram.
i.e. opposite angles are equal and sum of consecutive angles measure 180 degree.

One additional property of rhombus is that its diagonal bisect each other at 90 degrees.
i.e. ∠AOB = 180 degree

#### Angle property of Trapezium

In trapezium, one pair of sides are parallel to each other.

We know that when parallel sides are intersected by transversal then sum of interior angles measures 180 degree.

Hence,
∠A + ∠D = 180 degree
∠B + ∠C = 180 degree

### How to find missing angle in quadrilateral ?

If all the angle of quadrilateral is known except one, you can find the missing angle using angle sum property of quadrilateral.

Just subtract all the know angles with 360 degree and you will get the required value.

Let me show you the process with some examples;

Example 01
Find the value of missing angle x in below quadrilateral.

Solution
∠A = 90
∠D = 90
∠B = 120

Applying angle sum property of quadrilateral;
∠A + ∠B + ∠C+∠D = 360

90 + 120 + ∠C + 90 = 360

300 + ∠C = 360

∠C = 360 – 300

∠C = 60 degree

Hence, value of x is 60 degree.

Example 02
In the below figure, find the measure of angle x.

Solution
∠A = 120
∠D = 70
∠C = 65

Applying angle sum property of quadrilateral;
∠A + ∠B + ∠C+∠D = 360

120 + ∠B + 65 + 70 = 360

255 + ∠B = 360

∠B = 360 – 255

∠B = 105 degree

Hence, angle x measures 105 degree

Example 03
Given below is the parallelogram ABCD. Find the measure of all the angles.

Solution
In parallelogram, sum of adjacent angle measures 180 degrees.

∠B + ∠C = 180

x + 40 = 180

x = 140 degree

Also in parallelogram, opposite angles are equal in measure.
∠A = ∠C = 40 degree
∠D = ∠B = 140 degree

Example 04
Given below is the quadrilateral ABCD. Find the measure of all angles.

Solution
In quadrilateral, sum of all angles measures 360 degree.
∠A + ∠B + ∠C+∠D = 360

100 + (3x – 30) + (5x + 40) + 2x = 360

100 – 30 + 40 + (3x + 5x+ 2x) = 360

110 + 10x = 360

10x = 360 -110

10x = 250

x = 25

Now put the value of x in all the angles.
∠A = 100
∠B = 3x – 30 = 3(25) – 30 = 45
∠C = 5x + 40 = 5(25) + 40 = 165
∠D = 2x = 2 (25) = 50

Hence, we got the measure of all the angles.

Example 05
Given below is the quadrilateral ABCD. Find the measure of angle x.

Solution
First find interior ∠B of the given quadrilateral.

Since CB is a straight line, the sum of adjacent angle measure 180 degree.
∠B + 124 = 180

∠B = 180 – 124

∠B = 56 degree

Now apply angle sum property of quadrilateral.
∠A + ∠B + ∠C+∠D = 360

39 + 56 + 42 + x = 360

137 + x = 360

x = 360 – 137

x = 223

Hence, ∠x measures 223 degrees.