Complementary and Supplementary Angles

Complementary and Supplementary Angles are the two types of angles based on their measures. An angle that measures 900 is a Complementary angle and whose measure is 1800 is a Supplementary Angle.

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 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: An Angle measuring less than 90 degrees is called an acute Angle. So,  the value of an acute Angle varies from 0 to 90 degrees.

• Obtuse Angle: An Angle measuring more than 90 degrees but less than 180 degrees is called an obtuse Angle. So, the value of an obtuse Angle ranges from 90 degrees to 180 degrees.

• Right Angle: if an Angle measures exactly 90 degrees, then it is called a right Angle. If this right Angle is present in a triAngle, we call the triAngle a right-Angled triAngle. All the Angles in a square and a rectAngle are right Angles i.e, are equal to 90 degrees. 

• Reflex Angle: if an Angle measures less than 360 degrees but greater than 180 degrees, then it is known as a reflex Angle. So, the value of a reflex Angle ranges from 180 degrees to 360 degrees. 

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

• Complementary Angles: if two Angles are adjacent to each other or on the same line and  the sum of a pair of Angles is 90 degrees, we call them complementary Angles as they complement each other to make the sum equal to 90 degrees. 

• Supplementary Angles: even if two Angles are not adjacent to each other or are not on the same line, but they sum up to result in 180 degrees, we call them Supplementary Angles. 

• Adjacent Angles: if two Angles have a common side or an arm or a vortex i.e, point of origin of the ray, we call the Angles, adjacent Angles. Simply explained, two pizza slices that are side by side in a box are considered to be adjacent to each other. 

• Vertically Opposite Angles: vertically opposite Angles are equal to each other and are located opposite from a common vertex i.e, starting point or origin point. 

• Linear Pairs: linear pairs of Angles are adjacent to each other and have a common vertex and scissors is an excellent example to explain linear pair of Angles. 

In this session, we will be learning Complementary and Supplementary Angles in detail.

Complementary Angles Definition

When the sum of the measure of two Angles is 900, then the pair of Angles is said to be Complementary Angles. In Complementary Angles one Angle is a complement of the other making a sum of 900  or you can say forming a right Angle.

From the figure, we can say that ∠ABC + ∠CBD = 50 + 40 = 90o. This is an example of a Complementary Angles example.

But it is not necessary that the two Complementary Angles are always adjacent to each other. They can be different Angles, only their sum should be 900. 

Figure given below is a Complementary Angle example 27 + 63 = 90o.

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.

Supplementary Angles example

From the figure, we can say that  ∠1 +  ∠2 = 180°

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.

Supplementary Angles example

Properties of Supplementary Angles

• According to the Supplementary Angles definition 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.

Properties of Complementary Angles

• According to the supplementary Angles definition two Angles are said to be a Complementary Angle if the sum of their measures is 900.

• It is not necessary that a supplementary Angle will lie on the same line, they can be on different lines but should measure 900.

How to Solve Problems Related to Angles and Trigonometry?

• All the important Angles like acute Angle, obtuse Angle, right Angle, reflex Angle, complementary and supplementary Angles are explained above. Study their property thoroughly. 

• All you need to do is know the properties of an Angle and apply that to the given question. For example, let us assume that the question is to find out the other complementary Angle if one of the Angles measures 45 degrees. We know that the sum of the two Angles is 90 degrees if they are complementary Angles.

• Now, subtract the given value i.e, 45 degrees from 90 degrees to get the value of the other Angle: 90-45= 45 degrees. 

• A similar process can be applied to supplementary Angles but the sum value has to be changed to 180 degrees. 

How to Find Two Angles are Complementary?

If measures of two Angles are given. Add them. If the sum of the measures of these two Angles is 900, then the two Angles are Complementary Angles.

If the two Angles are given as Complementary Angles and if the measure of one Angle is given we can find the other Angle.

For Example:

∠A + ∠B = 90° and ∠A = 30°, then ∠B = ?

∠B = 90° - ∠A

∠B = 90° - 30°

∠B = 60°

The formula for finding a complementary Angle is 90 - x, where x is the measure of one of the Angles. 

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 = 80°, then ∠B = ?

∠B = 180 - ∠A

∠B = 180 - 80

∠B = 100°

The formula for finding a Supplementary Angle is 180° - x, where x is the measure of one of the Angles. 

Fun Facts

• In a right-angled triangle, the two non-right angles are complementary angles to each other.

• Complementary comes from the Latin word ‘completum’ meaning "completed" because the right angle is thought of as being a complete angle.

• The supplement comes from the Latin word ‘supplere’, to complete or "supply" what is needed.

Solved Examples

Example 1: Two angles are supplementary. One of these two angles is 1100 and finds the other angle.

Solution: One angle is given 1100

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 + 110° = 180°

x = 180° - 110°

x = 70°

So the other angle is 700

Example 2: The measure of an angle is 72°. What is the measure of a complementary angle?

Solution: Let x be the measure of the complementary angle.

Because x and 62° are complementary angles, we have

x + 62°  =  90°

x = 90°  - 72°

x = 18°

So, the measure of the complementary angle is 18°.

Example 3: Find the Supplement of the angle 1/3 of 240°.

Solution: 

Convert 1/3 of 240° 

That is, 1/3 x 240° = 80°

The sum of the measures of these two angles is 1800

x + 80° = 180°

x = 180° - 80°

x = 100° 

Therefore, Supplement of the angle 1/3 of 240° is 100°

Quiz Time

Supplementary Angles Examples

• The measure of an angle is 100°. What is the measure of a supplementary angle?

• Find the value of x

Complementary Angles Examples

• The measure of an angle is 22°. What is the measure of a complementary angle?

• Find the value of x