In this guide, we’ll take you step by step through everything you need to know about complementary angles.
We’ll start by breaking down what they are and how to spot them. Next, we’ll dive into the different types, like adjacent and non-adjacent angles, to show how they work in various scenarios. Together, we’ll solve some examples, and you’ll even get a chance to practice what you’ve learned!
Ready to explore complementary angles?
Let’s get started!
Meet Our Top-Rated Math Tutors in Blue Ash, OH
Complementary angles are a pair of angles whose sum equals 90°. Think of it like two puzzle pieces that perfectly fit together to form a right angle (a square corner).
They "complement" each other, meaning they complete each other to form that perfect 90° corner.
For example, a 30° angle and a 60°angle are complementary because when you add them together, they equal 90°.
Or, we can combine two 45° angles to form a 90° one—this makes the two 45° angles complementary.
Remember, the key is that the two angles must add up to 90° to be considered complementary.
As simple as that!
Complementary angles are two angles whose measures add up to 90°, while supplementary angles are two angles that sum up to 180°.
Think of complementary angles as forming a right angle, and supplementary angles as forming a straight line.
Why do we group angles like this?
These concepts aren’t just for solving geometry problems—they’re tools that help us understand how shapes and angles interact in both math and real life.
For example, architects and engineers use complementary and supplementary angles to design everything from buildings to bridges, ensuring structures are stable and functional. Artists and graphic designers also rely on these angle relationships to create balanced and visually appealing designs.
In math, understanding these terms helps us break down complex problems into simpler parts.
Whether you’re calculating unknown angles, solving puzzles, or even planning a piece of furniture, knowing how angles work together makes it easier to find solutions.
So, while they might seem like just definitions, complementary and supplementary angles are the building blocks for mastering geometry—and much more!
The Complementary Angles Theorem is a formalization of the idea behind complementary angles and gives us a mathematical tool to solve problems involving them.
The Complementary Angles Theorem states that:
If two angles are complementary, their measures add up to 90°.
Conversely, if the sum of two angles is 90°, then the two angles are complementary.
It not only tells us the key property of complementary angles but also allows us to apply that property logically in geometry and problem-solving.
The theorem helps us:
Find missing angles: If one angle is known, we can easily calculate the other by subtracting the given angle from 90°.
Prove relationships: In geometry, the theorem is often used in proofs and reasoning to establish angle relationships in triangles, intersecting lines, and more.
Simplify complex problems: By identifying complementary pairs, we can break down challenging problems into manageable steps.
Let’s say you’re given that one angle measures 60°, and you’re told it is complementary to another angle. The theorem allows us to calculate the unknown angle:
Unknown Angle = 90° - 30° = 60°
This means the second angle measures 30°, completing the complementary pair.
By formalizing the definition of complementary angles, the Complementary Angles Theorem gives us a powerful tool for understanding and solving problems involving angle relationships.
Complementary angles can be classified into different types based on their placement or how they interact with each other.
There are two types of complementary angles:
Adjacent Complementary Angles
Non-Adjacent Complementary Angles
Let’s explore them!
Adjacent complementary angles are two angles that share a common side and a common vertex, the point where the two angles meet. These angles sit next to each other and together form a perfect right angle.
Think of two puzzle pieces that connect seamlessly to create a 90° corner.
When placed together, they complete the picture of adjacent complementary angles in action.
Non-adjacent complementary angles are two angles that do not share a common side or vertex.
Although they are separate, their measures still add up to 90°, forming a right angle when combined.
Imagine two puzzle pieces placed anywhere on the board—apart from each other—but together they complete the shape of a right angle.
To see how these concepts are applied, let's examine some solved examples involving complementary angles.
What is the measure of an angle that is complementary to 82°?
82° + x = 90°
90° - 82° = 8°
The angle that is complementary to 82° is 8°.
Are these angles adjacent or non-adjacent?
Answer: Since these two angles do not share a common vertex, there are non-adjacent angles.
Two complementary angles have equal measures. What is the measure of each angle?
If a complementary angle adds up to 90° and the 2 sections are equal, let's set up an equation.
2x = 90°
x = 90° ÷ 2
x = 45°
Answer: Since each angle is equal, each angle measures 45°.
Find complementary angles: 2x
Two angles of equal degrees are complementary, or in other words, equal 90°.
2x = 90°
To find the degrees of these angles, we will divide 90° by 2.
x = 90° ÷ 2
x = 45°
Conclusion: The two complementary angles are 45° each.
Find complementary angles: x + 58°
We know that one angle is 58°. To find its complementary angle, all we have to do is subtract it from 90°.
x + 58° = 90°
x = 90° - 58°
x = 32°
Conclusion: The angle complementary to 58° is 32°.
Now it’s your turn to practice identifying and working with complementary angles!
Solve the problems below, and check your answers at the end to see how you did.
If one angle measures 63°, what is the measure of its complementary angle?
If one angle measures 25°, what is the measure of its complementary angle?
Two angles are complementary. One angle measures 48°. What is the measure of the other angle?
A carpenter is building a shelf. The shelf needs to be attached to the wall at a right angle. If the angle between the shelf and the wall is 32°, what is the angle between the wall and the support bracket?
Two angles are complementary. One angle is represented by 'x' and the other angle is represented by '2x'. Find the measure of each angle.
We’ve put together answers to some of the questions we often hear from students who are learning about complementary angles.
Yes! Two 45-degree angles are complementary because 45 + 45 = 90°.
No. A straight angle measures 180°, which is a supplementary angle. Complementary angles must add up to 90 degrees.
No. Complementary angles involve only two angles. If more angles are involved, their sum might still be 90°, but they wouldn’t be classified as complementary angles.
No, two obtuse angles cannot be complementary.
Complementary angles are two angles whose sum is 90 degrees, while an obtuse angle is any angle that measures greater than 90 degrees.
If you know the measure of one angle and it's complementary to another, you can find the missing angle by subtracting the known angle from 90°.
No, angles must have positive measures to qualify as complementary. Negative angles fall outside the standard definition of angles we use in this context.
Mathnasium tutors help students of all ages progress along their math journey.
At Mathnasium of Blue Ash, we're passionate about helping K-12 students of all skill levels succeed in math. Our specially trained tutors provide personalized in-center instruction and online support, guiding students through every step of their unique math journey.
Combining personalized learning plans and a proven teaching approach, we help them build a strong foundation and unlock their full potential.
Discover our approach to elementary school tutoring and middle school tutoring:
Mathnasium of Blue Ash follows personalized learning plans designed to put each student on their best path towards math excellence.
Solution: 90 degrees - 63 degrees = 27 degrees
Solution: 90 degrees - 25 degrees = 65 degrees
Solution: 90 degrees - 48 degrees = 42 degrees
Solution: 90 degrees - 32 degrees = 58 degrees
Solution:
x + 2x = 90
3x = 90
x = 90 ÷ 3
x = 30 degrees
2x = 60 degrees
(513) 984-1212
(513) 984-1212
