When two line segments or lines meet at a common point, at the point of intersection an angle is formed. When a ray is rotated about its end point, then the measure of its rotation in an anti-clockwise direction is the angle formed between its initial and final position.

In fig. 1 if the ray \(\small \overrightarrow{OP}\)Â is rotated in the direction of the ray \(\small \overrightarrow{OQ}\), then the measure of its rotation represents the angle formed by it. In this case, the measure of rotation that is the angle formed between the initial side and the terminal side is represented by ÆŸ.

Let us now have a look at complementary angles and supplementary angles.

## Complementary Angles

When the sum of two angles is 90Â°, then the angles are known as **complementary angles**. In other words, if two angles add up to form a right angle, then these angles are referred to as complementary angles.

In Fig. 2 given above, the measure of angle BOD is 60^{o} and angle AOD measures 30^{o}. On adding both of these angles we get a right angle, therefore âˆ BOD and âˆ AOD are complementary angles.

The following angles in Fig. 3 given below are complementary to each other as the measure of the sum of both the angles is 90^{o}. âˆ POQ and âˆ ABC are complementary and are called **complements** of each other.

** For example:** To find the complement of 2x + 52Â° , subtract given angle from 90 degrees.

90^{o}Â –Â (2x + 52^{o}) =Â 90^{o}Â – 2x – 52^{o}Â = -2x + 38^{o}Â

The complement of 2x + 52^{o} is 38^{o}Â – 2x.

**Important Observations:Â **

(a) Two right angles can not be complement of each other.

(b) Two obtuse angles can not be complement of each other.

(c) Two complementary angles are acute but vice versa is not possible.

## Supplementary Angles

When the sum of two angles is 180Â°, then the angles are known as supplementary angles. In other words, if two angles add up to form a straight angle, then these angles are referred to as **supplementary angles**.

In Fig. 4 given above, the measure of âˆ AOC is 60^{o} and âˆ AOB measures 120^{o}. On adding both of these angles we get a straight angle. Therefore, âˆ AOC and âˆ AOB are supplementary angles, and both of these angles are known as a supplement of each other.

### What is the difference between complementary and supplementary angles?

**Complementary angles**: Sum to 90 degrees

**Supplementary angles** : Sum to 180 degrees

**Supplementary angles**: Sum to 180 degrees

How could remember easily the difference between Complementary angle and supplementary angles?

- “
**C**“**Â letter ofÂ****C**omplementary stands for “**C**orner” (A right angle,Â \(90^{\circ}\) ) - “
**S**“**Â**letter ofÂ**S**upplementary stands for “**S**traight” ( a straight line, \(180^{\circ}\)))

## Examples

**Solved below example problems on supplementary and complementary angles.**

**Example 1:** Find the complement of 40 degrees.

**Solution:**Â As given angle is 40 degrees, then

Complement is 50 degrees.

We know that, Sum of Complementary angles =Â 90 degrees

So 40Â°Â + 50Â°Â = 90Â°

**Example 2:** Find the Supplement of the angle 1/3 of 210Â°.

**Solution:Â **

**Step 1:** Convert 1/3 of 210Â°Â

That is, 1/3 x 210Â° = 70Â°

**Step 2: **Supplement of 70Â° = 180Â° – 70Â° = 110Â°

Therefore, Supplement of the angle 1/3 of 210Â° is 110Â°

**Example 3:**Â The measure of two angles are (x + 25)Â° and (3x + 15)Â°. Find the value of x if angles are supplementary angles.Â

**Solution:Â **

We know that, **Sum of Supplementary angles =Â 180 degrees**

So,Â

(x + 25)Â° + (3x + 15)Â° = 180Â°Â

4x + 40Â°Â = 180Â°Â

4x = 140Â°Â

x = 35Â°Â Â

The value of x is 35 degrees.

**Example 4:** The difference between two complementary angles is 52Â°. Find both the angles.

**Solution:Â **

Let, First angle = m degrees then

Second angle =Â (90 – m)degrees Â Â {as per the definition of complementary angles}

Difference between angles = 52Â°Â

Now,

Â (90Â° – m) – m = 52Â°Â

90Â° – 2m = 52Â°Â

Â – 2m = 52Â° – 90Â°

-2m = -38Â°

m = 38Â°/2Â°

m = 19Â°

Again,Â Second angle = 90Â° – 19Â°Â = 71Â°Â

Therefore, the required angles are 19Â°, 71Â°.