Contents

- Definition
- Properties and Types of Complementary Angles
- Complement of an Angle
- Proof of Complementary Angles Theorem
- Difference Between Supplementary and Complementary Angles
- Complementary Angles And Their Importance in Mathematics
- Significance Of Right Angle Triangles
- Finding Angles Given That They Are Complementary

# Complementary Angle|Definition & Meaning

## Definition

When the sum of two angles is 90Â°, these two angles are called complementary angles.

**Properties and Types of Complementary Angles **

There are different properties of Complementary angles that are following

- If two angles sum is equal to 90 degrees, they are said to be complementary.
- They might be adjacent or not.
- Even though the total of three or more angles is 90 degrees, they cannot be complementary.
- When two angles are complementary, each angle is called the “complement” or “complementary” of the other.
- A right-angled triangle has two complimentary sharp angles.

There are two types of complementary angles i.e. Adjacent Complementary angles and non-adjacent Complementary angles.

Two angles whose sum is 90 and they have a **common vertex and arm** are known as adjacent complementary angles s shown in figure 3.

In figure 3 âˆ DAB is 18.12Â°and âˆ CAD is 72.24Â° the sum of angles is 90Â° .

If two complementary angles are **not adjacent** to each other they are **non-adjacent complementary angles **as illustrated in figure 4.

In figure 4, both the angles are non-adjacent, and their sum will be equal to 90Â°.

**Complement of an Angle**

Each angle is referred to as a “complement” of the other, and we know that the total of two complimentary angles equals 90 degrees. Therefore, an angle’s complement may be calculated by deducting it from 90 degrees. 90-xÂ° is the counterpart of xÂ°. Let’s calculate the angle’s complement, which is 57Â°. By deducting 57Â° from 90Â°, you may find its counterpart, which is 33Â°: 90Â° – 57Â°. Therefore, 33Â° is the complement of a 57Â° angle.

**Proof of Complementary Angles Theorem**

There are pairs of complementary angles that add up to 90Â°. Consider a diagram illustrated in Figure 2 to prove the complementary angle theorem.

We assume that <CBD is complementary to âˆ ABC and âˆ DBE.

As per the definition of Complementary angles, âˆ CBD +âˆ ABC=90Â°, and âˆ CBD +âˆ DBE=90Â°. Now we can say that âˆ CBD +âˆ ABC=âˆ CBD +âˆ DBE. Now we can see that âˆ ABC =âˆ DBE; hence theorem is proved.

**Difference Between Supplementary and Complementary Angles**

Two angles are called **supplementary** if their sum is 180Â° and two angles are **complementary** of each other if their sum is 90Â°. The supplement of an angle xÂ° is **180-xÂ°** whereas the complement of an angle xÂ° is **90-xÂ°.** Both the **supplementary angles** can be joined to form a straight angle, whereas the **complementary angles** can be joined to form a right angle.

**Complementary Angles And Their Importance in Mathematics**

Complementary angles are one of the most important parts of trigonometry. Trigonometry is one of the most significant areas of mathematics that has a wide range of applications. The study of the connection between the sides and angles of the right-angle triangle is essentially the focus of the field of mathematics known as **“trigonometry.”**

As complementary angles is the angles whose sum is 90, are used in vast fields like **astronomical research. Architects, surveyors, astronauts, physicists, engineers,** and even crime scene investigators use complementary angles in a variety of professions.

**Significance Of Right Angle Triangles**

More specifically, right-angled triangles with a 90Â° internal angle are the subject of trigonometry. We may use trigonometry to determine any missing or unknowable side lengths or angles in a triangle. All lengths in right-angled are not equal to each other. The side with 90Â° is known as **perpendicular,** the opposite side of the perpendicular is the **hypotenuse,** and the line segment on which they stand is the **base** of the triangle, as shown in figure 1.

As illustrated in Figure 1 we can see that the angle between the line segment h and f is 45Â° and between g and h is 45Â°, and the sum of the two angles is 90Â°, so both angles are the **complement **of each other.

The relationship between the **acute and right angle** of the right-angled triangle is defined by the trigonometric ratios, which are given below

$\sin(\theta)$= $\dfrac{\text{Perpendicular}}{\text{Hypotenuse}}$

$\cos(\theta)$= $\dfrac{\text{Base}}{\text{Hypotenuse}}$

$\tan(\theta)$= $\dfrac{\text{Perpendicular}}{\text{Base}}$

$\csc(\theta)$= $\dfrac{\text{Hypotenuse}}{\text{Perpendicular}}$

$\sec(\theta)$= $\dfrac{\text{Hypotenuse}}{\text{Base}}$

$\cot(\theta)$= $\dfrac{\text{Perpendicular}}{\text{Hypotenuse}}$

## Finding Angles Given That They Are Complementary

**Example 1**

Calculate the values of two complementary angles A and B if **A = (2x â€“ 18)Â° and B = (5x â€“ 52)Â°.**

**Solution**

âˆ A = (2x â€“ 18)Â° and âˆ B = (5x â€“ 52)Â°

We know that,

Sum of two complementary angles = 90Â°

âˆ A + âˆ B = 90Â°

(2x â€“ 18)Â° + (5x â€“ 52)Â° = 90Â°

7x â€“ 70Â° = 90Â°

7x = 90Â° + 70Â° = 160Â°

x = 160Â°/7 = 22.85Â°

Now,

âˆ A = (2 Ã— (22.857) â€“ 18) = 27.714Â°

âˆ B = (5 Ã— (22.857) â€“ 52) = 62.286Â°

Hence, âˆ A = 27.714Â° and âˆ B = 62.286Â°

**Example 2**

Find the value of X in given figure 5.

**Solution**

As illustrated in figure both angles x and 41.36Â° are complementary angles so their sum is 90Â°.

x + 41.36Â° = 90Â°

x = 90Â° – 41.36Â°

x = 48.64Â°

Therefore, the value of angle ‘x’ is 48.64Â°.

**Example 3**

Find x if the given angles illustrated in figure 6 are complementary angles.

**Solution**

As we know the sum of two complementary angles is 90Â°

$\dfrac{x}{2}$ + $\dfrac{x}{3}$ = 90Â°

$\dfrac{5x}{6}$ = 90Â°

x = 90Â° Ã— $\dfrac{6}{5}$ = 108Â°

Therefore, the value of x is 108Â°.

**Example 4**

Find the values of angles A and B such that âˆ A = (x – 25)Â° and âˆ B = (2x âˆ’ 25)Â° if both A and B are complementary of each other.

**Solution**

As âˆ A and âˆ B both are complementary so their sum is 90Â°.

âˆ A + âˆ B = 90Â°

(x – 25)Â° + (2x – 25)Â° = 90Â°

3x – 50Â° = 90Â°

3x = 140Â°

x = 46.6Â°

Thus, âˆ A = 46.6 – 25 = 21.6Â° and âˆ B = 2 (46.6) – 25 = 68.2Â°.

Therefore, âˆ A and âˆ B are 21.6Â°^{ }and 68.2Â°, respectively.

*All images/mathematical drawings were created with GeoGebra.*