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Complementary And Supplementary Angles

Understanding complementary angles supplementary angles is crucial in geometry, especially when dealing with angle relationships in triangles, polygons, and various geometric problems. These two types of angles form the foundation of angle-based reasoning in mathematics.

 

Table of Contents

 

Complementary Angles

What Is Complementary Angles?

Complementary angles are two angles whose measures add up exactly to 90 degrees.

If the sum of two angles is 90°, they are called complementary angles.

 

Definition:
Two angles are called complementary angles if the sum of their measures is 90°.

Formula:
If ∠A and ∠B are complementary, then:
∠A + ∠B = 90°

Example :
Two angles are complementary. One angle is twice the other.
Let one angle be x. Then the other = 2x
x + 2x = 90 → 3x = 90 → x = 30
So, the angles are 30° and 60°

 

Complementary Angles Formula

Formula:
If one angle is known (let’s say angle A), then the other angle (angle B) can be found using:

Complementary Angle = 90° – Known Angle

Mathematically:
If ∠A + ∠B = 90°, then
∠B = 90° – ∠A

 

Example:
If angle A = x and angle B = 2x, and they are complementary:
x + 2x = 90
3x = 90
x = 30°, then the other angle is 2x = 60°

 

Important Notes:  

  • The formula works only when two angles are complementary.

  •  Both angles must be less than 90°. 

  • This formula is often used in geometry, trigonometry, and triangle problems.

 

How to Do Complementary Angles  

Complementary angles are two angles that add up to 90 degrees.  

Steps to solve complementary angle problems:  

  1. Understand the concept.
    Two angles are complementary if their sum is 90°.  

  2. Use the formula:
    Complementary angle = 90° – given angle.  

  3. Solve the problem.

 

Example of Complementary Angles

Complementary angles are two angles whose sum is exactly 90 degrees.

Here are a few simple examples:

Example 1:
One angle is 35°.
Complementary angle = 90° – 35° = 55°
So, 35° and 55° are complementary angles.

Example 2:
One angle is 60°.
Complementary angle = 90° – 60° = 30°
So, 60° and 30° are complementary.

Example 3:
Two angles are complementary. One angle is twice the other.
Let the smaller angle be x.
Then the other is 2x.
x + 2x = 90
3x = 90 → x = 30
Other angle = 2x = 60
So, the complementary angles are 30° and 60°.

 

Example 4:
In a right triangle, the two non-right angles are always complementary.
If one angle is 40°, the other is 90° – 40° = 50°

 

Example 5:
If two angles are complementary and one angle is twice the other:
Let the smaller angle be x.
Then the other is 2x.
x + 2x = 90
3x = 90
x = 30
So the angles are 30° and 60°.

Final Check:
Always make sure the two angles add up to 90°.

 

Supplementary Angles

What Are Supplementary Angles?  

Supplementary angles are two angles that add up to exactly 180 degrees. This means if the sum of two angles is 180°, they are called supplementary angles.  

 

Definition:  

Two angles are supplementary if the sum of their measures is 180°.  

Mathematical Representation:  

If ∠A and ∠B are supplementary, then:  

 ∠A + ∠B = 180°

Example:
Two angles are in the ratio 2:3 and are supplementary.
Let the angles be 2x and 3x
2x + 3x = 180
5x = 180 → x = 36
So, the angles are:
2x = 72° and 3x = 108°
Hence, 72° and 108° are supplementary.

 

Supplementary Angles Formula

Supplementary angles are two angles whose measures add up to 180 degrees.

Formula:

If one angle is known, the other can be found using:

Supplementary Angle = 180° – Known Angle

Mathematical Representation:

If ∠A and ∠B are supplementary, then:
∠A + ∠B = 180°
So, if ∠A is known:
∠B = 180° – ∠A

 

How to Find Supplementary Angles

Supplementary angles are two angles whose measures add up to 180 degrees.

To find the missing angle when one is known, use the simple subtraction method.

Steps to Find a Supplementary Angle:

  1. Step 1: Understand that
    Angle A + Angle B = 180°
  2. Step 2: Use the formula
    Supplementary Angle = 180° – Known Angle
  3. Step 3: Solve the equation

 

Example of Supplementary Angles

Here are some examples to help you understand better:

Example 1:
One angle is 70°.
Find its supplementary angle.
Solution:
180° – 70° = 110°
So, 70° and 110° are supplementary angles.

 

Example 2:
Two angles are in the ratio 2:3 and are supplementary.
Solution:
Let the angles be 2x and 3x.
2x + 3x = 180°
5x = 180 → x = 36
2x = 72°, 3x = 108°
So, 72° and 108° are supplementary.

 

Example 3:
In a straight angle (straight line), if one angle is 125°, find the other.
Solution:
180° – 125° = 55°
So, the two supplementary angles are 125° and 55°.

 

Example 4:
If one angle is 90°, what is its supplement?
Solution:
180° – 90° = 90°
So, two right angles (90° + 90°) are also supplementary.

 

Real-Life Uses

Complementary angles:

  • Used in constructing right-angle triangles

  • Common in trigonometry

Supplementary angles:

  • Used in beam joining and road design

  • Found in angles on a straight line and linear pairs

Practice Questions

Q1. Find the complement of 38°
Q2. Find the supplement of 142°
Q3. Two angles are complementary. One is twice the other. Find both.
Q4. Two supplementary angles are in the ratio 1:2. Find the angles.
Q5. True or False: 80° and 100° are supplementary angles.
Q6. If one angle is 110°, find its supplementary angle.
Q7. Two supplementary angles differ by 40°. What are the angles?
Q8. Is 80° and 100° a pair of supplementary angles?
Q9. What is the supplementary angle of 75°?
Q10. If two angles are supplementary and one angle is 2x°, the other is (x + 30)°, find the value of x.

 

Fun Facts

  • "Complementary" comes from Latin complementum meaning "that which completes."

  • "Supplementary" comes from supplementum meaning "added."

  • If two complementary angles are equal, both are 45°.

  • If two supplementary angles are equal, both are 90°.

  • In right triangles, the non-right angles are always complementary.

Conclusion

Understanding complementary angles and supplementary angles is essential for mastering both basic and advanced geometry. These types of angles not only help in solving textbook problems but also matter in real-life situations like architecture, engineering, and trigonometry. 

By understanding what complementary angles and supplementary angles are, along with their formulas and practical uses, you build a solid foundation in mathematics.

 

Related Links

Angle Definition - Understand what an angle is, how it is formed by two rays meeting at a common endpoint, and explore real-life examples and basic angle terminology.

Types of Angles - Learn about various types of angles, such as acute, right, obtuse, straight, and reflex angles, with helpful illustrations and examples.

Interior Angles of a Polygon - Discover how to calculate the sum of interior angles in polygons, and understand the relationship between the number of sides and angle measurements.

 

Frequently Asked Questions On Complementary Angles and Supplementary Angles

1. What are complementary angles and supplementary angles?

Complementary angles are two angles whose measures add up to 90 degrees.
Example: 30° and 60° are complementary because 30 + 60 = 90°.

Supplementary angles are two angles whose measures add up to 180 degrees.
Example: 110° and 70° are supplementary because 110 + 70 = 180°.

 

2. What are the 7 types of angles with examples?

Here are the 7 types of angles with examples:

  • Acute Angle - less than 90°
    Example: 45°

  • Right Angle - exactly 90°
    Example: A corner of a square

  • Obtuse Angle - more than 90° but less than 180°
    Example: 120°

  • Straight Angle - exactly 180°
    Example: A straight line

  • Reflex Angle - more than 180° but less than 360°
    Example: 270°

  • Full Rotation Angle - exactly 360°
    Example: A full circle

  • Zero Angle  exactly 0°
    Example: Two rays on the same line in the same direction

3. Is a supplementary angle 90 or 180?

A supplementary angle adds up to 180 degrees, not 90.
Angles that sum to 90° are called complementary angles.

 

4. Is 20 and 70 complementary or supplementary?

20° + 70° = 90°
So, 20 and 70 are complementary angles because their sum is exactly 90°.

 

Keep practicing, explore more problems, and make learning angles both fun and easy with Orchids The International School.

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