Complementary Angles

If you've already been studying angles then you've likely learned about obtuse, right, acute, and straight angles. But did you know that all angles have a relationship with one another? We're going to take a look at one type of angle relationship: complementary angles.

What are complementary angles?

Complementary angles are two angles whose sum adds up to 90°. Two complementary angles can be adjacent. For example, in the figure below, $\angle 1$ and $\angle 2$ are adjacent:

Angles that are acute (greater than 0° but less than 90°) don't have to be adjacent and can still be complementary if their measures add up to 90°. For example, the figures below show two non-adjacent angles that are complementary because their measures $(35°+55°)$ add up to 90°.

Complementary angles theorem

According to the complementary angles theorem, two angles that complement the same angle must be congruent to one another.

Let's say you have three angles:

$\angle 1$ has a measure of 51° and is complementary to $\angle 2$ and $\angle 3$

$\angle 2$ has a measure of 39° and is complementary to $\angle 1$

$\angle 3$ has an unknown measure and is complementary to $\angle 1$

You don't know the measure of $\angle 3$ , but you know that $\angle 2$ and $\angle 3$ are both complementary to $\angle 1$ . In this case, $\angle 3$ is congruent to $\angle 2$ and must also have a measure of 39°.

Complementary angles vs. supplementary angles

It's common to discuss supplementary angles in the same conversation with complementary angles because they both represent an angle relationship. While the measures of two complementary angles add up to 90°, the measures of two supplementary angles add up to 180°.

For example, the figure below shows two supplementary angles ( $\angle 1$ and $\angle 2$ ). These angles also make up a linear pair:

Practice questions on complementary angles

a. $\angle Q$ has a measure of 37° and $\angle R$ has a measure of 51°. Are they complementary angles?

To check if they are complementary, we need to see if their measures add up to 90°:

$35°+55°=88°$

No, because their measures add up to 88°.

b. True or false: Adjacent angles cannot be complementary angles.

False. Adjacent angles can be complementary if their measures add up to 90°.

c. $\angle 1$ has a measure of 44° and $\angle 2$ has a measure of 46°. Are they complementary angles?

To check if they are complementary, we need to see if their measures add up to 90°:

$44°+46°=90°$

Yes, because their measures add up to 90°.

d. Let's say you have three angles: $\angle A$ is 22°, $\angle B$ is 13°, and $\angle C$ is 68°. Which two of these three angles are complementary?

To check for complementary angles, we need to see which pair adds up to 90°:

$22°+13°=35°$

$22°+68°=90°$

$13°+68°=81°$

$\angle A$ and $\angle C$ are complementary.

e. $\angle X$ is 14°, $\angle Y$ is 76°, and $\angle Z$ is unknown. If $\angle X$ and $\angle Z$ are both complementary to ∠Y, what is the measure of $\angle Z$ ?

Since $\angle X$ and $\angle Z$ are complementary to $\angle Y$ , their measures add up to 90°:

$90°+76°=14°$

The measure of $\angle Z$ is 14°.

f. If $\angle S$ is 125° and $\angle T$ is 55°, what is their angle relationship?

To check if they are supplementary, we need to see if their measures add up to 180°:

$125°+55°=180°$

They are supplementary angles.

Topics related to the Complementary Angles

Supplementary Angles

Adjacent Angles

Reference Angle

Flashcards covering the Complementary Angles

Basic Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Complementary Angles

Common Core: High School - Geometry Diagnostic Tests

Intermediate Geometry Diagnostic Tests

Get help learning about complementary angles

Complementary angles can be a complex concept for first-time learners. Your student might need to decide whether two angles are complementary, or they could have to determine the measure of an angle that is complementary to one angle and congruent to another. Whether your student is having trouble grasping complementary angles or they feel good about their knowledge and would like help with an assignment or test preparations, they could benefit from studying alongside a private tutor. Contact the Educational Directors at Varsity Tutors today to learn more about the perks of tutoring.