 # Supplementary Angles

## What is a Supplementary Angle?

When the sum of the measure of two angles is 1800, then the pair of angles is said to be supplementary angles. We have studied basic geometrical terms. We know that two lines having a common vertex form an angle. Various parts of an angle are vertex, arms, interior, and exterior of angles. When two lines intersect they form four angles at the point of intersection. An angle is denoted by the symbol ∠.

From the figure, ∠ ABC is an angle. B is the point of intersection called the vertex and AB and BC are the sides of the angle. Angles are commonly measured in terms of degree.

Angles are classified according to their sizes as follows

• Acute angle

• Obtuse angle

• Right angle

• Reflex angle

There are more types of angles on the basis of the pairs of angles. They are as follows

• Complementary angles

• Supplementary angles

• Vertically opposite angles

• Linear Pairs

In this session, we will be learning what is a supplementary angle?

Supplementary meaning is to complete or supply what is needed.

Supplementary Angles Definition

When the sum of the measure of two angles is 1800, then the pair of angles is said to be supplementary angles. Here the supplementary meaning is one angle is supplemented to another angle to make a sum of 1800.

From the figure, we can say that  ∠1 +  ∠2 = 1800

But it is not necessary that the two supplementary angles are always adjacent to each other. They can be different angles, only their sum should be 1800

From the figure, ∠3  and  ∠4 are not adjacent but they make a supplementary angle if their sum is 1800.

### Properties of Supplementary Angles

• Two angles are said to be a supplementary angle if the sum of their measures is 1800.

• It is not necessary that a supplementary angle will lie on the same line, they can be on different lines but should measure 1800.

• In the supplementary angles if one is an acute angle then the other is an obtuse angle.

• In a supplementary angle if one angle is 900 the other angle will also be 900.

### Supplementary Theorem

• If a ray stands on a line, then the sum of two adjacent angles so formed is 1800.

•  If two angles let ∠A and ∠B are both supplementary to a third angle be ∠C, then ∠A and ∠B are congruent.

If ∠A + ∠C = 180°

∠B + ∠C = 180° then ∠A = ∠B.

### How to Find two Angles are Supplementary?

If measures of two angles are given. Add them. If the sum of the measures of these two angles is 1800, then the two angles are supplementary angles.

If the two angles are given as supplementary angles and if the measure of one angle is given we can find the other angle.

For example : ∠A + ∠B = 180° and ∠A = 800 , then ∠B = ?

∠B = 180 - ∠A

∠B = 180 - 80

∠B = 1000

Supplementary Angles can be classified as adjacent supplementary angles and non- adjacent supplementary angles.

Supplementary angles that share one common arm are called adjacent supplementary angles.

Supplementary angles that do not share any common arm and are totally two different angles then they are called non-adjacent supplementary angles.

### Solved Examples

Example 1: Two angles are supplementary. One of these two angles is 500 find the other angle.

Solution :

Let the other angle be x

Given that the two angles are supplementary we have,

The sum of the measures of these two angles is 1800

x + 50 = 1800

x = 180 - 50

x = 1300

So the other angle is 1300

Example 2: Two angles are supplementary and one of them is 71°.What is the size of the other angle?

Solution: Let ‘a’ be the size of the other angle.

Given that the two angles are supplementary

Then by supplementary angles definition, we have

The sum of the measures of these two angles is 1800

a + 71 = 180

a = 180 - 71

a = 1090

Therefore the size of the other angle is 1090.

### Quiz Time

1. Two angles are supplementary and one of them is twice as big as the other. What is the size of the smaller angle?

2. Two angles are supplementary and one of them is 137°. What is the size of the other angle?

1. What is the Difference Between Supplementary Angles and Complementary Angles?

•  If the sum of the measure of two angles is 1800 then the angles are said to be supplementary angles, whereas if the sum of measures of two angles is 900 then the angles are called complementary angles.

•  Supplementary angles together form a straight line whereas complementary angles together form a right angle.

• Two angles ∠A and ∠B  are supplementary angles if ∠A + ∠B = 1800, while two angles ∠A and ∠B are complementary if ∠A + ∠B = 900

• Example of supplementary angle

∠1 + ∠2 = 1800

Example of Complementary angle

∠1 + ∠2 = 900

2. Can two Obtuse Angles be Supplementary?

Answer: No two obtuse angles cannot be supplementary because according to the definition of obtuse angle an angle is called an obtuse angle if its measure is greater than 900. And if two obtuse angles are added their measure will be more than 1800 which contradicts the definition of the supplementary angle. Two angles are supplementary if the sum of the measures of their ankle is 1800.

For example, consider two obtuse angles of measures 1000 and 1100 then the sum will be 100 + 110 =  2100

Hence the result of the sum of two obtuse angles is 2100 which is greater than the supplementary angles.

So we can say that two obtuse angles cannot be supplementary.

3. Can Two Acute Angles be Supplementary?

Answer: No, two acute angles cannot be supplementary angles because according to the definition of acute angle an angle is called an acute angle if its measure is less than 900. And if two acute angles are added then their sum will be less than 1800 which contradicts the definition of the supplementary angle. The sum of the measures of the supplementary angle is 1800