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# Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In other words, if you have two angles and you add their measures together, and the result is 90 degrees, then the angles are complementary.

• Complementary Angles
• Formula for Complementary Angles
• Solved Examples
• FAQs

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## Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In other words, if you have two angles and you add their measures together, and the result is 90 degrees, then the angles are complementary.

For example, if one angle measures 30 degrees, then the other angle that is complementary to it would measure 60 degrees. If one angle measures 45 degrees, then the other angle that is complementary to it would measure 45 degrees as well. Complementary angles can be formed in many different ways, such as by two intersecting perpendicular lines, or by a right triangle. In a right triangle, one of the angles measures 90 degrees, and the other two angles are complementary.

Complementary angles are important in many areas of mathematics, including geometry and trigonometry. They can also be used in real-world applications, such as in surveying, navigation, and construction, where angles need to be measured and adjusted to ensure accuracy and precision.

## Formula for Complementary Angles

The formula for complementary angles is quite simple. If two angles are complementary, their measures add up to 90 degrees. In mathematical notation, if we let angle A and angle B be two complementary angles, we can write:

A + B = 90 degrees

This formula can be used to find the measure of one angle when the measure of the other angle is known. For example, if we know that angle A measures 30 degrees, we can find the measure of its complementary angle B as follows:

A + B = 90 degrees

30 + B = 90

B = 90 – 30

B = 60

So the measure of angle B is 60 degrees, since it is complementary to angle A which measures 30 degrees.

## Complementary Angles in Triangles

In a triangle, the three angles add up to 180 degrees. If one of the angles in a triangle is a right angle (i.e., measures 90 degrees), then the other two angles must be complementary. This is because:

A right angle measures 90 degrees

The sum of the other two angles in the triangle must be 180 – 90 = 90 degrees

Therefore, the other two angles must be complementary.

For example, in a right triangle, one angle measures 90 degrees and the other two angles are complementary to each other.

If one of the complementary angles measures 30 degrees, then the other complementary angle must measure 60 degrees, since:

30 degrees + 60 degrees = 90 degrees (which is the measure of the right angle)

Similarly, if one of the complementary angles in a triangle measures x degrees, then the other complementary angle must measure 90 – x degrees.

Complementary angles in triangles are important in geometry and trigonometry, and can be used to solve problems involving the angles and sides of triangles.

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## Solved Examples of Complementary Angles

Example 1: If one angle in a right triangle measures 30 degrees, what is the measure of the other acute angle?

Solution:

Since one angle in a right triangle is a right angle (measures 90 degrees), the other two angles are complementary. Let’s call the other acute angle x. We know that:

30 degrees + x = 90 degrees (the sum of the two complementary angles)

x = 90 – 30 = 60 degrees

Therefore, the measure of the other acute angle in the right triangle is 60 degrees.

Example 2: If one angle of a triangle measures 40 degrees, what is the measure of the other acute angle?

Solution:

Since the three angles of a triangle add up to 180 degrees, and one of the angles is 90 degrees or less (i.e., acute), the other two angles must be complementary. Let’s call the other acute angle x. We know that:

40 degrees + x = 90 degrees (the sum of the two complementary angles)

x = 90 – 40 = 50 degrees

Therefore, the measure of the other acute angle in the triangle is 50 degrees.

Example 3: The measure of one angle of a triangle is 30 degrees less than its complementary angle. Find the measures of the two angles.

Solution:

Let’s call one of the angles x (the smaller one), and the other angle y (the larger one). We know that:

x + y = 90 degrees (since the angles are complementary)

y = x + 30 (since one angle is 30 degrees less than its complementary angle)

Substituting the second equation into the first equation, we get:

x + (x + 30) = 90 degrees

2x + 30 = 90 degrees

2x = 60 degrees

x = 30 degrees

So the measure of the smaller angle is 30 degrees. Substituting this value back into the second equation, we get:

y = x + 30 = 30 + 30 = 60 degrees

So the measure of the larger angle is 60 degrees.

## Complementary Angles FAQS

##### What is the sum of two complementary angles?

The sum of two complementary angles is always 90 degrees.

##### Can two obtuse angles be complementary?

No, two obtuse angles cannot be complementary. Since an obtuse angle measures greater than 90 degrees, it is not possible for two obtuse angles to add up to 90 degrees.

##### Are all right angles complementary?

No, right angles are not complementary to each other. A right angle measures exactly 90 degrees, which means that it cannot be combined with any other angle to make 90 degrees.

##### How can complementary angles be used in real life?

Complementary angles can be used in a variety of real-life applications, such as in architecture, engineering, and construction, where angles need to be measured and adjusted to ensure accuracy and precision. They can also be used in navigation and surveying to determine the direction and position of objects and landmarks.

##### How can I remember the difference between complementary and supplementary angles?

One way to remember the difference is to think of the words themselves. Complementary angles “complement” each other to make 90 degrees, while supplementary angles “supplement” each other to make 180 degrees. Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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