JUMP TO TOPIC

- 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 90°, 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 ∠DAB is 18.12°and ∠CAD is 72.24° the sum of angles is 90° .

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 90°.

**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-x° is the counterpart of x°. Let’s calculate the angle’s complement, which is 57°. By deducting 57° from 90°, you may find its counterpart, which is 33°: 90° – 57°. Therefore, 33° is the complement of a 57° angle.

**Proof of Complementary Angles Theorem**

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

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

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

**Difference Between Supplementary and Complementary Angles**

Two angles are called **supplementary** if their sum is 180° and two angles are **complementary** of each other if their sum is 90°. The supplement of an angle x° is **180-x°** whereas the complement of an angle x° is **90-x°.** 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 90° 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 90° 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 45° and between g and h is 45°, and the sum of the two angles is 90°, 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 – 18)° and B = (5x – 52)°.**

**Solution**

∠A = (2x – 18)° and ∠B = (5x – 52)°

We know that,

Sum of two complementary angles = 90°

∠A + ∠B = 90°

(2x – 18)° + (5x – 52)° = 90°

7x – 70° = 90°

7x = 90° + 70° = 160°

x = 160°/7 = 22.85°

Now,

∠A = (2 × (22.857) – 18) = 27.714°

∠B = (5 × (22.857) – 52) = 62.286°

Hence, ∠A = 27.714° and ∠B = 62.286°

**Example 2**

Find the value of X in given figure 5.

**Solution**

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

x + 41.36° = 90°

x = 90° – 41.36°

x = 48.64°

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

**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 90°

$\dfrac{x}{2}$ + $\dfrac{x}{3}$ = 90°

$\dfrac{5x}{6}$ = 90°

x = 90° × $\dfrac{6}{5}$ = 108°

Therefore, the value of x is 108°.

**Example 4**

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

**Solution**

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

∠A + ∠B = 90°

(x – 25)° + (2x – 25)° = 90°

3x – 50° = 90°

3x = 140°

x = 46.6°

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

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

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