**Supplementary angles and complementary angles** are defined with respect to the addition of two angles. If the sum of two angles is 180 degrees then they are said to be supplementary angles, which forms a linear angle together. Whereas if the sum of two angles is 90 degrees, then they are said to be complementary angles and they form a right angle together.

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 endpoint, 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.

## 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. Here we say, that the two angles complement each other. Suppose if one angle is x then the other angle will 90^{o} – x. Hence, we use these complementary angles for trigonometry ratios, where on ratio complement another ratio by 90 degrees such as;

- sin (90Â°- A) = cos A and cos (90Â°- A) = sin A
- tan (90Â°- A) = cot A and cot (90Â°- A) = tan A
- sec (90Â°- A) = csc A and csc (90Â°- A) = sec A

Hence, you can see here the trigonometric ratio of the angles gets changed if they complement each other.

In the above figure, 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.

**Points to Remember**

(a) Two right angles cannot complement each other.

(b) Two obtuse angles cannot complement 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 those angles are referred to as **supplementary angles**. These two angles form a linear angle, where if one angle is x, then the other the angle is 180 – x. The linearity here proves that the properties of the angles remain the same. Take the examples of trigonometric ratios such as;

- Sin (180 – A) = Sin A
- Cos (180 – A) = – Cos A (quadrant is changed)
- Tan (180 – A) = – Tan A

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.

### 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, \(180^{\circ}\)))

## Supplementary and Complementary Examples

The example problems on supplementary and complementary angles are given below:

**Example 1:**

Find the complement of 40 degrees.

**Solution:**Â

As the 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°.