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How to find the Complement of an Angle?

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In geometry, complementary angles can be defined as those angles whose sum is 90 degrees. For example, 39° and 51° are complementary angles, as the sum of 39° and 51° is 90°. If the sum of two angles is a right angle, then we can say that they are complementary angles. But what is an angle? In geometry, an angle is referred to as the space formed between two rays when they are joined together by a common point called a vertex.  If θ is an angle, then (90° – θ) is the complementary angle of θ.

 

For two angles to be complementary, their sum must be 90 degrees, i.e., the two angles must be acute.  If θ is an angle, then (90° – θ) is the complementary angle of θ.

 

Types of Complementary angles

Two angles are said to be complementary if their sum is 90°. In geometry, there are two types of complementary angles, i.e., adjacent complementary angles and non-adjacent complementary angles.

Adjacent complementary angles: Two complementary angles having a common vertex and a common arm are called adjacent complementary angles. 

 

From the given figure, we can say that ∠QEF and ∠DEQ are adjacent angles, as both angles share the common vertex “E” and the common arm “EQ”. Since ∠QEF + ∠DEQ = 17° + 73° = 90°, ∠QEF and ∠DEQ are also complementary angles. Therefore, the two given angles are adjacent complementary angles. 

Non-adjacent Complementary Angles: Two angles are said to be non-adjacent angles if they don’t share a common vertex and a common arm. Non-adjacent complementary angles are complementary angles that are not adjacent to each other.

 

From the given figure, we can say that ∠XYZ and ∠ABC are non-adjacent angles, as both angles do not share a common vertex and a common arm.  âˆ XYZ and ∠ABC are also complementary angles since their sum is 90°, i.e., ∠XYZ + ∠ABC = 57° + 33° = 90°. Therefore, the given two are non-adjacent complementary angles. 

Complementary angles theorem

The complementary angles theorem states that “If two angles are a complement to any third angle, then the first two angles are congruent to each other.”

 

Proof:

Let us assume that ∠COB is complementary to ∠BOA and ∠DOC.

From the definition of the complementary angles we get, 

∠COB + ∠BOA = 90°         ————— (1)

∠COB + ∠DOC = 90°         ————— (2)

From equations (1) and (2) we can say that,

∠COB + ∠BOA = ∠COB + ∠DOC 

⇒ ∠COB + ∠BOA – ∠COB – ∠DOC = 0

⇒ ∠BOA – ∠DOC = 0

⇒ ∠BOA = ∠DOC

Hence, the theorem is proved.

Properties of complementary angles

Let us discuss some properties of complementary angles.

  1. A pair of angles are said to be complementary if they add up to 90°.
  2. The two complementary angles can either be adjacent or non-adjacent.
  3. An angle is said to be the complement of another angle if the sum of both angles is 90°.
  4. Even if the sum of three or more angles is 90°, they can not be complementary.
  5. The two complementary angles are acute.

Finding the complement of an angle

To find the complement of an angle, we need to subtract the given angle from 90°, as we know that the sum of two complementary angles is 90°. If θ is the given angle, then (90° – θ) is the complement of θ. 

For example, calculate the complement of 17°.

We know that the sum of two complementary angles is 90°.

As a result, the complement of 17° is (90° – 17°) = 73°.

Hence, the complement of 17° is 73°.

Difference between complementary and supplementary angles

Complementary angles

 Supplementary angles 

If the sum of a pair of angles is 90°, then they are said to be complementary.

 If the sum of a pair of angles is 180°, then they are said to be supplementary.

(90° – θ) is the complement of an angle θ.

(180° – θ) is the supplement of an angle θ.

If a pair of complementary are joined together, then they form a right angle.

If a pair of supplementary are joined together, then they form a straight line.

For two angles to be complementary, their sum must be 90 degrees, i.e., the two angles must be acute.

In two supplementary angles, one angle is acute and the other is obtuse, or both of them may be right angles.

Solved Problems

Problem 1: Calculate the values of the two complementary angles, A and B, if A = (2x – 18)° and B = (5x – 52)°. 

Solution:

Given data,

∠A = (2x – 18)° and ∠B = (5x – 52)°

We know that,

Sum of two complementary angles = 90°

∠A + ∠B = 90°

⇒ (2x – 18)° + (5x – 52)° = 90°

⇒ 7x – 70° = 90°

⇒ 7x = 90° + 70° = 160°

⇒ x = 160°/7 = 22.85°

Now, 

∠A = (2 × (22.857) – 18) = 27.714°

∠B = (5 × (22.857) – 52) = 62.286‬°

Hence, ∠A = 27.714° and ∠B = 62.286‬°.

Problem 2: Determine the value of x if (5x/3) and (x/6) are complementary angles.

Solution:

Given data,

(5x/3) and (x/6) are complementary angles.

We know that,

Sum of two complementary angles = 90°

⇒ (5x/3) + (x/6) = 90°

⇒ (10x + x)/6 = 90°

⇒ 11x = 90° × 6 = 540°

⇒ x = 540°/11 = 49.09°

Hence, the value of x = 49.09°.

Problem 3: Find the value of x in the figure shown below.

 

Solution:

From the given figure we can observe that x and 54° are complementary angles, i.e., the sum of x and 54° is 90°.

⇒ x + 54° = 90°

⇒ x = 90° – 54° = 36°

Hence, the value of x is 36°.

Problem 4: Find the value of y and the measure of angles in the given figure.

 

Solution:

From the given figure, we can observe that (2y – 15)° and (3y – 25)° are complementary angles, i.e., the sum of (2y – 15)° and (3y – 25)° is 90°.

⇒ (2y – 15)° + (3y – 25)° = 90°

⇒ (5y – 40)° = 90°

⇒ 5y = 90° + 40° = 130°

⇒ y = 130°/5 = 26°

Now, (2y – 15)° = ( 2 × 26 – 15) = 37°

(3y – 25)° = (3 × 26 – 15) = 53°

Hence, the value of y is 26° and the complementary angles are 37° and 53°.

Problem 5: Determine the value of x and the measure of complementary angles in the figure shown below.

 

Solution:

Given that, (x – 3)° and (2x – 7)° are complementary angles, i.e., the sum of (x – 3)° and (2x – 7)° is 90°.

⇒ (x – 3)° + (2x – 7)° = 90°

⇒ (3x – 10)° = 90°

⇒ 3x = 90° + 10° = 100°

⇒ x = 100°/3 = 33.34°

Now, (x – 3)° = (33.333- 3)° = 30.333° = 30.33°

(2x – 7)° = (2 × (33.333) – 7)° = 59.666° = 59.67°

Hence, the value of x is 33.333° and the three complementary angles are 30.33° and 59.67°.



Last Updated : 27 Jun, 2022
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