Complementary Angles
The word "complementary" came from two Latin words "Complere" and "Plere". "Complere" means "complete", whereas "Plere" means "fill". So "complementary" means "something that completes and brings perfection." And so are complementary angles, a pair of two angles that sum up to 90 degrees, forming a right angle.
A slice of bread is rectangular in shape, but when it is divided into two pieces by cutting along the diagonal, two right triangles are formed, each with a pair of complementary angles. In this lesson, we will explore the world of complementary angles.
What are Complementary Angles?
The complementary and supplementary of the two angles is decided by the sum of their measurement. If the sum of the two angles is equal to the measurement of a right angle then the pair of angles is said to be complementary angles.
Complementary Angles Definition:
Two angles are said to be complementary angles if they add up to 90 degrees. In other words, when complementary angles are put together, they form a right angle (90 degrees). Angle 1 and angle 2 are complementary if the sum of both the angles is equal to 90 degrees (angle 1+ angle 2 = 90°) and thus, angle 1 and angle 2 are called complements of each other.
Complementary Angles Example:
In the figure given below, 60° + 30° = 90°. Hence, from the "Definition of Complementary Angles", these two angles are complementary. Each angle among the complementary angles is called the "complement" of the other angle. Here,
 60° is the complement of 30°
 30° is the complement of 60°
Adjacent and NonAdjacent Complementary Angles
If the sum of two angles is equal to the measurement of a right angle then the pair of angles is known as the complementary angle. There are two types of complementary angles in geometry as given below:
 Adjacent Complementary Angles
 Nonadjacent Complementary Angles
Adjacent Complementary Angles: Two complementary angles with a common vertex and a common arm are called adjacent complementary angles. In the figure given below, ∠COB and ∠AOB are adjacent angles as they have a common vertex "O" and a common arm "OB". They also add up to 90 degrees, that is ∠COB + ∠AOB = 70°+20° = 90°. Thus, these two angles are adjacent complementary angles.
Nonadjacent Complementary Angles: Two complementary angles that are NOT adjacent are said to be nonadjacent complementary angles. In the figure given below, ∠ABC and ∠PQR are nonadjacent angles as they neither have a common vertex nor a common arm. Also, they add up to 90 degrees that is, ∠ABC + ∠PQR = 50° + 40° = 90°. Thus, these two angles are nonadjacent complementary angles. When nonadjacent complementary angles are put together, they form a right angle.
How to find Complement of an Angle?
We know that the sum of two complementary angles is 90 degrees and each of them is said to be a "complement" of each other. Thus, the complement of an angle is found by subtracting it from 90 degrees. The complement of x° is 90x°. Let's find the complement of the angle 57°. The complement of 57° is obtained by subtracting it from 90°. 90°  57° = 33°. Thus, the complement of 57° angle is 33°.
Properties of Complementary Angles
Now we have already learned about the types of complementary angles. Let's have a look at some important properties of complementary angles. The properties of complementary angles are given below.
 Two angles are said to be complementary if they add up to 90 degrees.
 Two complementary angles can be either adjacent or nonadjacent.
 Three or more angles cannot be complementary even if their sum is 90 degrees.
 If two angles are complementary, each angle is called "complement" or "complement angle" of the other angle.
 Two acute angles of a rightangled triangle are complementary.
Complementary Angles v/s Supplementary Angles
The supplementary and complementary angles are angles that exist in pairs, summing up to 180 and 90 degrees, and have numerous realtime applications, most common being the crossroads. Let's have a look at the difference between them.
The supplementary vs complementary angles table:
Supplementary Angles  Complementary Angles 

A pair of angle are said to be supplementary if their sum is 180 degrees.  A pair of angle are said to be complementary if their sum is 90 degrees. 
Supplement of an angle x° is (180  x)°  The complement of an angle x° is (90  x)° 
Here is a short trick for you to understand the difference between supplementary angles and complementary angles.
 "S" is for "Supplementary" and "S" is for "Straight." Hence, you can remember that two "Supplementary" angles when put together form a "Straight" angle.
 "C" is for "Complementary" and "C" is for "Corner." Hence, you can remember that two "Complementary" angles when put together form a "Corner (right)" angle.
Complementary Angle Theorem (with Illustration)
If the sum of two angles is 90 degrees, then we say that they are complementary. Each of the complement angles is acute and positive. Let's study the complementary angle theorem with its proof. The complementary angle theorem states, "If two angles are complementary to the same angle, then they are congruent to each other".
Proof of Complementary Angle Theorem
We know that complementary angles exist in pairs and sum upto 90 degrees. Consider the following figure and prove the complementary angle theorem.
 Let us assume that ∠POQ is complementary to ∠AOP and ∠BOQ.
 Now as per the definition of complementary angles, ∠POQ + ∠AOP = 90° and ∠POQ + ∠BOQ =90° .
 From the above two equations, we can say that "∠POQ + ∠AOP = ∠POQ + ∠BOQ".
 Now subtract '∠POQ' from both sides, ∠AOP = ∠BOQ
 Hence, the theorem is proved.
Solved Examples

Example 1:
Find the angle x in the following figure.
Solution
In the given figure, x and 62° are complementary as they form a right angle. Hence, their sum is 90°
x + 62° = 90°
x = 90°  62°
x = 28°
Therefore, the angle 'x' is 28°.

Example 2: Find the values of Angle A and Angle B if Angle A and Angle B are complementary such that Angle A = (3x  25)° and Angle B = (6x − 65)°
Solution
Since ∠A and ∠B are complementary, their sum is 90°, ∠A + ∠B = 90° , which means (3x  25) + (6x  65) = 90° , 9x  90 = 90° , 9x = 180° , x = 20°. Thus, ∠A = 3 (20)  25 = 35° and ∠B = 6 (20)  65 = 55°
Therefore, ∠A and ∠B are 35°^{ }and 55° respectively.

Example 3: Find the value of x if the following two angles are complementary.
Solution
Since the given two angles are complementary, their sum is 90°. This means x/2 + x/3 = 90°, 5x/6 = 90°, x = 90° × 6/5 = 108°
Therefore, the value of x is 108°

Example 4: Two angles are complementary. The measure of the larger angle is 5 degrees more than 4 times the measure of the smaller angle. What is the measure of the larger angle in degrees?
Solution
Let us assume that the two complementary angles are x (larger) and y (smaller). By the given information, x = 4y + 5. Since the two angles are complementary, their sum is 90°, x + y = 90°
(4y + 5) + y = 90°
x = 4y + 5, 5y + 5 = 90°, 5y = 85°, y = 17°
Thus, the larger angle is, x = 4 (17) + 5 = 73°
Therefore, the measurement of the largest angle is 73°.
FAQs on Complementary Angles
How do you Find a Complementary Angle?
If the sum of two angles is 90 degrees, then we say that they are complementary. Thus, the complement of an angle is obtained by subtracting it from 90. For example, the complement of 40° is 90°^{ } 50° = 40°
What is the Complementary Angle of a 40 Degrees Angle?
The complement of an angle is obtained by subtracting it from 90 degrees. Thus, the complement of 40° is 90°^{ } 50° = 40°
How do you Find the Value of x in Complementary Angles?
If two angles in terms of x are given to be complementary, we just set their sum equal to 90 degrees and solve the resultant equation. You can refer to "Example 2" and "Example 3" under the "Solved Examples" section of this page.
How do you Find the Ratio of Two Complementary Angles?
The ratio of two complementary angles is found just like how we find the ratio of any two quantities. For example, the ratio of the complementary angles 40° and 50° is \(\dfrac{40}{50}= \dfrac{4}{5}\)
Complementary Angles Form what type of angle?
Two angles form a pair of complementary angles when their sum is 90°. So, the pair of complementary angles form a right angle.
Are Supplementary and Complementary Angles the Same?
No, supplementary and complementary angles are not the same. Two angles form a pair of complementary angles when their sum is 90° whereas two angles form a pair of supplementary angles when their sum is 180°
What is the Opposite of Complementary?
The opposite of "complementary" in math can be "supplementary". Two angles form a pair of supplementary angles when their sum is 180°, whereas the two angles form a pair of complementary angles when their sum is 90°.
Can Two Right Angles be Complementary Angles?
The measure of a right angle is 90°. The sum of two right angles will be 180° which is greater than 90°. So, two right angles can never be complementary angles.