Complementary Angles

Two acute angles are said to be complementary angles if the sum of the angles equals 90° i.e., complementary angles are those angles whose sum adds up to 90°. In other words, we can say that the sum of complementary angles is 90°.

Complementary Angles are not just a term that is used in geometry as this also has some real-life applications. One such example is slicing a rectangular-shaped bread along the diagonal. We will get two right triangles, each with a pair of complementary angles. In this article, we will be learning about the definition of complementary angles, types of complementary angles, properties, and how to find the complement of an angle.

What are Complementary Angles?

In geometry, two angles are known as complementary angles if their sum is a right angle i.e. 90° (90 degrees). In other words, when two angles add to 90°, they are called complementary angles.

For example, if ∠A = 40°, then its complementary angle is ∠A = 50° and 40° + 50° = 90°. In this case, 40° and 50° are complements of each other. i.e.

• 40° is the complement of 50° , and
• 50° is the complement of 40°.

Complementary Angles Definition

If the sum of two acute angles equals to the measurement of a right angle then, the pair of angles is known as complementary angles.

Sum of Complementary Angles

If ∠A and ∠B are complementary angles, then their sum will be 90° i.e., ∠A + ∠B = 90°

Complementary Angles Examples

In the above figure, 59° and 31° are complementary angles as the sum of 59° and 31° equals 90° or we can that 59° is the complement of 31° and 31° is the complement of 59°.

Let us take another example, 25° and 65° are also complementary angles as the sum of 25° and 65° equals 90°. Here, we can say that 25° and 65° are complements of each other.

Complementary Angles in Real Life

Some of the examples in real life of Complementary Angles are illustrated in following collage.

Types of Complementary Angles

Complementary angles are the pair of angles whose sum is equal to the measurement of a right angle. There are two types of complementary angles which are given below:

• Adjacent Complementary Angles
• Non-Adjacent Complementary Angles

Adjacent Complementary Angles

Adjacent means close or near something and complementary means 90°. So, if a pair of complementary angles have common vertex and a common arm, they are known as adjacent complementary angles.

In the figure shown above, ∠XOZ and ∠ZOY have a common vertex “O” and a common arm “OZ”. Also, they add up to 90°, i.e., ∠XOZ + ∠ZOY = 30° + 60° = 90°. Hence, these two angles are adjacent complementary angles.

Non-Adjacent Complementary Angles

Non – adjacent means something which is not close and complementary means 90°. So, if a pair of complementary angles does not have a common vertex or a common arm, they are known as non – adjacent complementary angles.

In the figure given above, ∠GFE and ∠JIH does not have a common vertex “or a common arm. But, they add up to 90° i.e., ∠GFE + ∠JIH = 22° + 68° = 90°. Hence, these two angles are non – adjacent complementary angles.

Read More about Types of Angles.

Properties of Complementary Angles

• Complementary angles are the pair of angles whose sum of the measures is equal to 90°.
• If two angles are known as complementary, we call each angle as “complement” or “complement angle” of the other angle.
• There are two types of complementary angles – adjacent and non-adjacent.
• Three or more angles can not be considered as complementary angles even if their sum is 90°. Complementary angles appear in pairs always.
• The pair of complementary angles are always acute, but not all pairs of acute angles are complementary.
• Two right angles or two obtuse angles can not form a complementary pair.

How to Find Complement of an Angle?

Complementary angles are the pairs of angles whose sum equals 90°. Also, each angle in the pair is considered to be the “complement” of the other. Suppose the angle given is y°. To find it’s complement, you subtract its value from 90° i.e., we can use the formula given below.

Complement of y° = (90 – y)°

For example, to determine the complement of an angle measuring 59°, you subtract 59° from 90°, yielding a complement of 33°. So, the complement of a 59° is 33°.

Complementary Angle Theorem

If the sum of pair of angles equals 90°, then they are said to be complementary. Each of the complement angles is acute and has positive measure. Now, let us study about the complementary angles theorem along with its proof.

According to the Complementary Angle Theorem, “if two angles are complementary to the same angle, then they are congruent or equal to each other.”

Proof of Complementary Angles Theorem

We know that complementary angles always exist in pairs and add up to 90°. Let us consider the following figure.

Let us assume that ∠POQ is complementary to ∠AOP and ∠QOR.

According to the definition of complementary angles, ∠POQ + ∠AOP = 90° and ∠POQ + ∠QOR = 90°.

From the above two equations, we can say that “∠POQ + ∠AOP = ∠POQ + ∠QOR”.

On subtracting ‘∠POQ’ from both sides, we get ∠AOP = ∠QOR.

Thus, the complementary angle theorem is proved.

Complementary Angles in Trigonometry

Let us observe the following figure.

In △ABC, ∠A + ∠C = 90°

So, we can say that ∠A and ∠C are complementary angles.

Read More about Trigonometry.

Trigonometric Ratios of Complementary Angles

Let us consider △ABC again.

If ∠A = θ, then ∠C = 90° – θ

and 

So, sin θ = cos (90° – θ)

Similarly, the following also holds true for θ

• cos θ = sin (90° – θ)
• tan θ = cot (90° – θ)
• cot θ = tan (90° – θ)
• cosec θ = sec (90° – θ)
• sec θ = cosec (90° – θ)

Suppose we have to find the value of sin 52° – cos 38°.

We know that sin θ = cos (90° – θ)

sin 52° = cos (90° – 52°) = cos 38°

sin 52° – cos 38° = cos 38° – cos 38° = 0

Read More about Trigonometric Ratios.

Complementary Angles vs Supplementary Angles

Complementary and Supplementary Angles sum up to specific measures, namely 90° and 180°, respectively. These angles can be either adjacent or non-adjacent. Complementary angles collectively make up a right angle, which is 90°, while supplementary angles combine to form a straight angle, equal to 180°. The comparison between complementary and supplementary angles is summarised in the table below:

Complementary Angles	Supplementary Angles
Two angles are known as complementary angles if the sum of angles is 90°.	Two angles are known as supplementary angles if the sum of angles is 180°.
Both complementary angles are acute and have positive measures.	Both supplementary angles can neither be acute or obtuse. Either both of them can be 90° each or one is acute angle whereas other is obtuse angle.
Complement of angle x° = (90 – x)°	Supplement of angle x° = (180 – x)°

Read More,

• Alternate Exterior Angles
• Lines and Angles
• Measuring Angles

Solved Examples on Complementary Angles

Example 1: Are 46° and 44° complementary angles? Give reason.

Solution:

Given two angles i.e., 46° and 44° .

We know that two angles are said to be complementary if their sum is 90°.

Since 46° + 44° = 90°

Thus, 46° and 44° are complementary angles.

Example 2: Find the complement of 61°.

Solution:

We know that the complement of an angle measuring x° = (90 – x)°.

Here, x° = 61°

Complement of 61° = (90 – 61)° = 29°

Thus, the complement of 61° = 29°

Example 3: Find the value of sec 37° cosec 53° – tan 37° cot 53°.

Solution:

sec 37° cosec 53° – tan 53° cot 37°

We know that cot θ = tan (90° – θ) and cosec θ = sec (90° – θ)

cot 53° = tan (90° – 53°) and cosec 53° = sec (90° – 53°)

cot 53° = tan 37° and cosec 53° = sec 37°

On substituting the values, we get

sec 37° sec 37° – tan 37° tan 37°

= sec²37° – tan²37° = 1

Hence, sec 37° cosec 53° – tan 37° cot 53° = 1

Example 4: Find the value of x, if (x + 3)° and (x – 5)° form complementary angles.

Solution:

We know that two angles are said to be complementary if their sum is 90°.

So, (x + 3)° + (x – 5)° = 90°

(x + x + 3 – 5)° = 90°

(2x – 2)° = 90°

2(x – 1)° = 90°

(x – 1)° = 45°

x = (45 + 1)°

x = 46°

Hence, the value of x is = 46°.

Example 5: If sin 2A = cos (A – 27°), where A is an acute angle, find A.

Solution:

sin 2A = cos (A – 27°)

We know that sin θ = cos (90° – θ)

sin 2A = cos (90° – 2A)

On substituting the value, we get

cos (90° – 2A) = cos (A – 27°)

90° – 2A = A – 27°

90° + 27° = 3A

A=\frac{117^{\circ}}{3}

A = 39°

Practice Problems on Complementary Angles

Problem 1: Find the complement of 16°.

Problem 2: Find the value of cos 23° cosec 67°.

Problem 3: Two complementary angles are equal. Find the measure of each angle.

Problem 4: If tan 6θ = cot 9θ, then what is the value of θ?

Problem 5: Find the value of sin 15° sin 25° sin 45° sec 65° sec 75°.

Complementary Angles – FAQs

1. What does Complementary Angle Means?

Complementary angles are two angles whose measures add up to 90°. In other words, if angle A is complementary to angle B, then A + B = 90°.

2. What is the Formula for Complementary Angles?

In trigonometry, the formulas for complementary angles are given below:

• sin θ = cos (90° – θ)
• cos θ = sin (90° – θ)
• tan θ = cot (90° – θ)
• cot θ = tan (90° – θ)
• cosec θ = sec (90° – θ)
• sec θ = cosec (90° – θ)

3. How do you Find the Complementary Angles?

If the sum of two angles = 90°, then we can say that the angles are complementary. It means complementary angles add up to 90°. Hence, we can obtain the complement of angle by finding the difference between 90° and the angle given. For example, the complement of 50° is 90° – 50° = 40°

4. Can Two Right Angles form Complementary Angles?

The sum of two right angles will be equal to 180° whereas complementary angles are the angles whose sum is equal to 180°. So, two right angles can not form complementary angles.

5. What is Complementary Angles and Supplementary Angles?

A pair of angles is said to be complementary if they sum up to 90° i.e., if their sum equals 90°. On the other hand, a pair of angles is said to be supplementary if they add up to 180° i.e., if their sum equals 180°.

6. Can two Obtuse Angles form Complementary Angles?

Obtuse angle is an angle whose measure is more than 90° but less than 180°. If we add two angles more than 90°, then we will get a reflex angle. So, two obtuse angles can not form complementary angles.

7. What are Two equal Complementary Angles?

Two equal complementary angles are the angles which are equal and also form complementary angles i.e., they add up to 90°. The measure of each equal complementary angle is 45°.

8. Are Complementary Angles Positive?

Yes, complementary angles are positive. They can never be negative or zero.

9. Can Vertically Opposite Angles form a pair of Complementary Angles?

Yes, two vertically opposite angles can form a pair of complementary angles. It is only possible when each vertical angle measures 45° as vertically opposite angles are equal.

10. Can three Angles be Complementary?

No, three angles can never be complementary angles because complementary angles form a pair of angles whose sum is equal to 90 degree.

11. Which two Complementary Angles differ by 20°?

Two complementary angles which differ by 20° are 35° and 55° .