In geometry, identifying geometric properties of shapes such as lines and angles is an essential skill that you need to acquire to understand more profound concepts such as determining and finding the measures of angles.
This time, we will familiarize ourselves with another unique pair of angles called complementary angles.
Are you thrilled to learn about this type of angle pair? Get ready as we are going to go indepth in exploring the world of complementary angles!
…but first, let’s define an angle before we go deeper into understanding complementary angles.
Complementary Angles
So, what exactly are complementary angles?
Complementary angles are a pair of angles that, when combined, sum up to 90 degrees.
You can imagine them as two puzzle pieces that fit together to form a 90degree angle. When talking about complementary angles, it is important to keep in mind that they always appear in twos.
We can say that one angle is the complement of the other or that one angle is complementary to another.
If an angle measures 90 degrees, it is called a right angle. Since right angles do not require another piece of the puzzle to complete the 90degree angle, they do not have complements and cannot also be called a complement of their own.
Three or more angles cannot also be called complementary angles even if their measures add up to 90 degrees because, by definition, complementary angles are always a pair.
As a rule, the measures of complementary angles are always positive.
Notice that complementary angles are always acute because each of the complements must measure less than 90 degrees to add up to 90 degrees.
In the illustration below, the sum of
Similarly, the sum of two
You might have heard of or used the word “complimentary” before, and you might think that it has nothing to do about adding up to 90 degrees.
Notice that the spelling of the word “complimentary” is spelled slightly different from what we are talking about here, which is “complementary.” So, be careful with the spelling, okay?
Did you know that…
The word “complementary” is derived from two Latin words, “complere,” which means “complete,” and “plere,” which means “fill?” The actual meaning of the word “complementary” is “the combination of objects or things in such a way that they complete each other or enhance the qualities of one another.”
Isn’t it cool to know the origin of the word complementary?
Types of Complementary Angles
There are two types of complementary angles – the adjacent complementary angles and the nonadjacent complementary angles.
Let’s take a look at how these two types differ from one another.
Adjacent Complementary Angles
These are complementary angles that share a common vertex and a common side. Below are illustrations of adjacent complementary angles.
Nonadjacent Complementary Angles
These are complementary angles that are not adjacent to each other. Below are illustrations of nonadjacent complementary angles.
Finding the Complement of an Angle
At some point in time, we may encounter situations that may require us to find the complement of an angle. Are you thrilled to know how to do it?
Here’s a simple method of determining the complement of a specific angle!
Since complementary angles add up to 90 degrees, we can determine the complement of an angle by subtracting the angle’s measure from
For example, let us determine the complement of an angle whose measure is
Therefore, the complement of an angle whose measure is
What a simple thing to do! Now you know how to find the complement of a specific angle.
Complementary angles also have some interesting characteristics and features that we’ll look at in the next section.
Properties of Complementary Angles
To fully understand complementary angles, you must keep in mind the following concepts that include some important characteristics and features:
 Complementary angles are those whose sum of the measures is equal to 90 degrees.
 If two angles are complementary, we call each angle “complement” or “complement angle” of the other angle.
 Complementary angles can be either adjacent or nonadjacent.
 Even if the sum of three or more angles is 90 degrees, they cannot be considered complementary angles. Complementary angles always appear in pairs.
 Two right angles or two obtuse angles can’t complement one another.
 A pair of complementary angles is always acute, but not all pairs of acute angles are complementary.
Complementary Angle Theorem
The complementary angle theorem states that if two angles are complementary to the same angle, then they are congruent.
But how did this theorem exist? Let’s find out by proving.
Suppose we have three angles, namely
Given that
By Subtraction Property of Equality, we can rewrite the previous equations as
Since the lefthand side of both equations is equal then,
Given that the measures of
Therefore, we have proven the theorem.
Solving Problems Involving Complementary Angles
Let us now try to solve some problems related to what we have just learned about complementary angles.
Problem 1
Find the measure of angle y in the figure.
In the given figure, y and
So, we can write it as
Therefore, angle y measures
Now, let’s solve another problem.
Problem 2
Suppose
This looks challenging but, it is really quite simple to solve.
Since
Given that
Substituting the value of
Let us now substitute the value of x in
Therefore, the measure of
Complementary Angles in Real World
For example, when you slice a rectangularshaped bread along the diagonal, you will have two right triangles, each with a pair of complementary angles.
More so, when the clock’s hour hand and minute hand form a right angle, say 3 o’clock, whenever the secondhand passes between 12 and 3, it forms complementary angles.
Another fine thing about complementary angles is that it is used to create designs. Having the two boards meet with perfect 45degree cuts leaves the corner at 90 degrees. These diagonal cuts can be seen in the corners of picture frames, which add a better aesthetic feature.
Can you tell other things that exhibit complementary angles?
Take a Quiz
Now that you have come to know a lot of things about complementary angles, are you ready to practice and apply your knowledge about it?
Which of the following pairs of angles are complementary?

and and and and
1.
Two complementary angles have measures in the ratio of 1 to 5. Find the measurement of the larger angle.
2.
If an angle is six more than twice the measure of its complement, what is the measure of the larger angle?
2.
Based on the given figure, what is the measure of angle x?
Frequently Asked Questions
Complementary angles are pairs of angles whose measures add up to 90 degrees.
The sum of the measure of complementary angles is always
No. Only angles whose measure is less than 90 degrees have a complement.
No. Complementary angles always appear in pairs.
Simply subtract the measure of the given angle from