Students in the US first encounter angles in 3rd or 4th grade, starting with right angles, the perfectly square corners you see on desks, books, doors, and windows.
Complementary angles build directly on that and usually come into focus around 4th to 7th grade, depending on the curriculum.
If you're just learning about complementary angles, need a refresher, or are preparing for an exam, this is the guide for you.
Today, Mathnasium tutors walk you through the definition and the different types of complementary angles with worked examples, practice problems, and answers to the most common questions students have about complementary angles.
Complementary angles are two angles whose sum equals 90°.
Each angle in a complementary pair is called the complement of the other.
To find a missing complementary angle, subtract the known angle from 90°.
Complementary angles can be adjacent, sharing a common side and vertex, or non-adjacent, sitting apart from each other.
Two 45° angles are the only pair of equal complementary angles.
Complementary angles do not have to be next to each other to be complementary.
A right angle can be made up of any two complementary angles.
An angle is a figure formed by two rays that share a common endpoint. The shared endpoint is a vertex, and the two rays are the sides of the angle.
We measure angles in degrees, written with the symbol °. Degrees tell us how wide or narrow the opening between the two rays is.
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Complementary angles are a pair of angles whose sum equals 90°.
They "complement" each other, which means they complete each other to form a right angle. Just like two puzzle pieces that fit together perfectly to form a square corner.
For example, a 30° angle and a 60° angle are complementary because if we add them together, they equal 90°.
Two 45° angles work the same way.
Complementary angles are two angles whose measures add up to 90°, while supplementary angles are two angles that sum up to 180°.
For instance, if we add two angles, 136° and 44°, we get a supplementary angle.
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 go beyond just 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 we are 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.
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The Complementary Angles Theorem gives us a precise, reliable way to work with complementary angles in geometry and problem-solving.
The theorem states two things:
If two angles are complementary, their measures add up to 90°.
It works the other way too: if two angles add up to 90°, they are complementary.
We use the Complementary Angles Theorem in three main ways to:
Find missing angles: If we know one angle, we can find its complement by subtracting it from 90°. For example, if one angle measures 35°, we subtract 35° from 90° and get 55°. No complex equation needed.
Prove relationships: In geometry, we often use this theorem in proofs to establish how angles relate to each other, especially in triangles and intersecting lines, which are shapes and figures students encounter regularly in class.
Break down complex problems: Many geometry problems look intimidating at first glance. If we spot a complementary pair inside a larger figure, we’ll solve one part of the problem at a time, which makes the whole thing more manageable.
Let's see the theorem in action. Say we know one angle measures 60°, and we are told it is complementary to another angle. We subtract 60° from 90°:
Unknown Angle (x) = 90° - 60° = 30°
The two complementary angles are 60° and 30°.
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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
Adjacent complementary angles are two angles that always 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.
Non-adjacent complementary angles are two angles that add up to 90° but do not meet both requirements for being adjacent. They may share a vertex, or a side, or nothing at all. They never share both.
In this image, we have two examples of non-adjacent angles.
The angles on the left share a side only, but not the vertex. That means they meet only one of the two requirements for adjacent angles.
Two angles on the right do not share anything, and that makes them non-adjacent as well.
Let’s go through a few solved examples of complementary angles together!
What is the measure of an angle that is complementary to 82°?
To calculate this, we need to set up the equation like this:
82° + x = 90°
x = 90° - 82° = 8°
The angle that is complementary to 82° is 8°.
What does the image tell us? Are the angles adjacent or non-adjacent?
We can see that these two angles share a side. However, they do not share a common vertex. So, they are non-adjacent angles.
Let’s check this image. 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°
Since each angle is equal, our final answer is that each angle measures 45°.
Now, let’s find complementary angles: 2x = 90°
Two angles of equal degrees are complementary, or in other words, equal 90°. To find the degrees of these angles, we will divide 90° by 2.
x = 90° ÷ 2
x = 45°
We reached the result that the two complementary angles are 45° each.
Shall we now find complementary angles in this equation: x + 58° = 90°?
We know that one angle is 58°. To find its complementary angle, we need to subtract it from 90°.
x = 90° - 58°
x = 32°
The angle we found complementary to 58° is 32°.
Ready to practice what we’ve covered? Give these challenges a try on your own and check your answers at the bottom of the guide.
If one angle measures 63°, what is the measure of its complementary angle?
If one angle is 25°, what is the complementary angle?
Two angles are complementary. One angle measures 48°. What is the measure of the other angle?
A father rests a ladder flat on the ground with one end against the wall. After a long day of work, he manages to lift it to 32° before he needs his son's help. Together, they need to raise the ladder all the way to 90° so it stands flat against the wall. How many more degrees do they need to lift it?
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 put together answers to some of the questions we hear most often from students learning about complementary angles.
No. Complementary angles involve only two angles. If more angles are involved, their sum might still be 90°, but we would not classify them as complementary angles.
No. An obtuse angle already measures more than 90° on its own, so two obtuse angles can never add up to exactly 90°.
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No. Angles need positive measures to qualify as complementary. Negative angles fall outside the standard definition we use in geometry.
Mathnasium's specially trained tutors help students work through complementary angles in a caring and fun group environment.
Mathnasium is the only math-only learning center helping K-12 students of all skill levels learn and master any math topic, including complementary angles.
For us, mastery goes beyond getting the right answer. Mastery means understanding why the answer is correct well enough to apply the same reasoning to a problem they have never seen before.
To help students reach that level of understanding, we use the Mathnasium Method™, a proprietary teaching approach. Unlike a one-size-fits-all curriculum, our approach is designed around each student's individual needs and learning style.
Each student begins with a diagnostic assessment that reveals their strengths and knowledge gaps. Those insights inform a personalized learning plan tailored to their goals.
With the plan in place, our tutors deliver face-to-face instruction in a warm and supportive setting.
To make math make sense, we use natural language and a combination of verbal, visual, mental, tactile, and written techniques.
We break each concept down into manageable steps, showing them both the how and the why behind the answer. This helps them develop the problem-solving tools and critical thinking they can use in math and beyond.
Fun is a core part of how we work. Our activities are often game-based, students earn rewards, and every bit of their progress is celebrated. This keeps them truly engaged and their confidence growing with every session.
And the results speak for themselves:
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With more than 1,100 learning centers, Mathnasium brings top-rated math instruction close to home.
For families in and around Blue Ash, OH, Mathnasium of Blue Ash is a trusted local center with experience helping students transform how they think and feel about math. Our center has been recognized for the quality of instruction we bring to the community:
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If you worked through the practice problems, here are the answers:
63° + x = 90° → x = 90° - 63° → x = 27°
25° + x = 90°→ x = 90° - 25° → x = 65°
48° + x = 90° → x = 90° - 48° → x = 42°
32° + x = 90° → x = 90° - 32° → x = 58°
Father and son need to lift the ladder for 58° more.
x + 2x = 90° → 3x = 90° → x = 90° ÷ 3 → x = 30° → 2x = 60°
The angle represented by ’x’ is 30° , while the angle ‘2x’ is 60°.
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