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In geometry, complementary angles are defined as two angles whose sum is 90 degrees. In other words, two angles that add up to 90 degrees are known as complementary angles. For example, if angle A is 20 degrees, then its complement angle B would be 70 degrees because 20 degrees + 70 degrees = 90 degrees. In this case, 20 degrees and 70 degrees are complements of each other. i.e.,
- 20 degrees is the complement of 70 degrees and
- 70 degrees is the complement of 20 degrees.
Let us learn more about it in this article.
What are Complementary Angles?
The complement and supplement of the two angles are 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 their sum is equal 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 (i.e. angle 1 + angle 2 = 90°) and thus, angle 1 and angle 2 are called complements of each other.
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°.
Types of 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
- Non-adjacent 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.
Non-adjacent Complementary Angles: Two complementary angles that are NOT adjacent are said to be non-adjacent complementary angles. In the figure given below, ∠ABC and ∠PQR are non-adjacent 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 non-adjacent complementary angles. When non-adjacent 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 the other. Thus, the complement of an angle is found by subtracting it from 90 degrees. The complement of x° is 90-x°. Let's find the complement of the angle 57°. The complement of 57° is obtained by subtracting it from 90°, i.e. 90° - 57° = 33°. Thus, the complement of 57° angle is 33°. Here is an interesting example about complement of an angle.
Example: Find the angle which is equal to its complement.
Solution: Let the required angle be x. Then its complement is (90 - x). It is given that:
angle = its complement
x = 90 - x
2x = 90
x = 45
Thus, the angle which is equal to its complement is 45 degrees.
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.
- They can be either adjacent or non-adjacent.
- Three or more angles cannot be complementary even if their sum is 90 degrees.
- If two angles are complementary, each angle is called the "complement" or "complement angle" of the other angle.
- Two acute angles of a right-angled triangle are complementary.
Complementary Angles and Supplementary Angles
The complementary and supplementary angles are those that add up to 90 degrees and 180 degrees respectively. They can either be adjacent or non-adjacent. When complementary angles can be considered as two parts of a right angle, the supplementary angles are the two parts of a straight angle or a 180-degree angle. The difference between complementary angles and supplementary angles are given in the table below:
|Supplementary Angles||Complementary Angles|
|A pair of angles are said to be supplementary if their sum is 180 degrees.||A pair of angles 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)°.|
|They can be joined together to form a straight angle.||They can be joined together to form a right angle.|
Here is a short trick for you to understand complementary angles vs supplementary angles.
- "S" for "Supplementary" = "Straight" angle
- "C" for "Complementary" = "Corner" (right) angle
Complementary Angles Theorem
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 angles 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 Angles Theorem:
We know that complementary angles exist in pairs and sum up to 90 degrees. Consider the following figure and prove the complementary angle theorem.
- Let us assume that ∠POQ is complementary to ∠AOP and ∠QOR.
- Now as per the definition of complementary angles, ∠POQ + ∠AOP = 90° and ∠POQ + ∠QOR = 90°.
- From the above two equations, we can say that "∠POQ + ∠AOP = ∠POQ + ∠QOR".
- Now subtract '∠POQ' from both sides, ∠AOP = ∠QOR.
- Hence, the theorem is proved.
☛ Related Articles:
Check out the following important articles to know more about complementary angles in math.
Complementary Angles Examples
Example 1: Find the angle x in the following figure.
In the given figure, x and 62° are complementary angles as they form a right angle. Hence, their sum is 90°.
x + 62° = 90°
x = 90° - 62°
x = 28°
Answer: The value of angle 'x' is 28°.
Example 2: Find the values of two complementary angles A and B such that ∠A = (3x - 25)° and ∠B = (6x − 65)°.
Since ∠A and ∠B are complementary, their sum is 90°.
⇒ ∠A + ∠B = 90°
⇒ (3x - 25)° + (6x - 65)° = 90°
⇒ 9x - 90° = 90°
⇒ 9x = 180°
⇒ x = 20°.
Thus, ∠A = 3 (20) - 25 = 35° and ∠B = 6 (20) - 65 = 55°.
Answer: ∠A and ∠B are 35° and 55° respectively.
Example 3: Find the value of x if the following two angles are complementary.
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°
Answer: The value of x is 108°.
FAQs on Complementary Angles
What is the Meaning of Complementary Angles?
In geometry, two angles are said to be complementary angles if they add up to 90 degrees. If ∠1 and ∠2 are complementary angles, then ∠1 + ∠2 = 90°.
What do Complementary Angles Add up to?
Two complementary angles always add up to 90 degrees. If ∠A and ∠B are complementary angles, it implies that:
- ∠A + ∠B = 90°.
- ∠A is the complement of ∠B.
- ∠B is the complement of ∠A.
What is a Pair of Complementary Angles?
Two angles form a pair of complementary angles when their sum is 90°. So, the pair of complementary angles form a right angle.
Find an Angle Whose Complement is Equal to its Supplement.
Let the angle that is to be found be x. Then its complement and supplement are 90 - x and 180 - x respectively. Then they said 90 - x = 180 - x. Solving this, 90 = 180, which is never true. Hence, there does not exist angle whose complement is same as its supplement.
How to Find Complementary Angles?
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° - 40° = 50°.
What is the Sum of Two Complementary Angles?
The sum of two complementary angles is always 90 degrees. Hence, if X and Y are complementary, this implies that ∠X + ∠Y = 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.
What is the Difference Between Complementary and Supplementary Angles?
The supplementary angles are those whose sum is 180 degrees while the sum of two complementary angles is 90 degrees. Two supplementary angles form a straight angle and two complementary angles form a right angle.
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. If one angle is given as x°, then the measurement of another angle is 90° - x°.
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°.