Complementary Angles
Look at the corner of your notebook. It forms a perfect right angle, which is 90 degrees. Now imagine splitting that right angle into two smaller angles. Those two smaller angles that add up to 90 degrees are called complementary angles.
Complementary angles appear everywhere around us. When a ladder leans against a wall, the angle between the ladder and the ground and the angle between the ladder and the wall are complementary. When you fold a square piece of paper along the diagonal, the two angles at the fold are complementary.
The word "complementary" comes from the Latin word "complere" meaning "to complete." Two complementary angles complete a right angle (90 degrees) together.
In this chapter, you will learn the definition of complementary angles, how to find the complement of a given angle, and how to solve problems involving complementary angles. The ideas here are simple, but they form the building blocks for more advanced geometry in higher classes.
What is Complementary Angles?
Definition: Two angles are called complementary angles if the sum of their measures is 90 degrees.
Each angle is called the complement of the other.
Angle 1 + Angle 2 = 90 degrees
Key Points about Complementary Angles:
- The two angles do NOT need to be next to each other (adjacent). They can be anywhere. As long as their sum is 90 degrees, they are complementary.
- Each of the two angles must be an acute angle (less than 90 degrees). If one angle were 90 degrees or more, the sum would exceed 90 degrees.
- A right angle (90 degrees) does NOT have a complement, because the other angle would have to be 0 degrees.
- The complement of an angle A is (90 - A) degrees.
Examples of Complementary Pairs:
| Angle 1 | Angle 2 | Sum | Complementary? |
|---|---|---|---|
| 30 degrees | 60 degrees | 90 degrees | Yes |
| 45 degrees | 45 degrees | 90 degrees | Yes |
| 10 degrees | 80 degrees | 90 degrees | Yes |
| 55 degrees | 35 degrees | 90 degrees | Yes |
| 50 degrees | 50 degrees | 100 degrees | No |
| 40 degrees | 60 degrees | 100 degrees | No |
Complementary Angles Formula
Formula to Find the Complement of an Angle:
Complement of angle A = 90 degrees - A
Where:
- A is the given angle
- A must be less than 90 degrees (an acute angle) for the complement to exist
Using Complementary Angles in Equations:
If two complementary angles are given in terms of a variable, set up the equation:
Angle 1 + Angle 2 = 90 degrees
Solve for the variable, then find each angle.
Types and Properties
Complementary angles can appear in different forms:
Adjacent Complementary Angles
Two angles that are next to each other (share a common side and vertex) and together form a right angle. You can see this when a right angle is split into two parts by a line.
Non-Adjacent Complementary Angles
Two angles that are NOT next to each other but still add up to 90 degrees. For example, one angle is in one corner of the room and another angle is drawn on paper. If their measures add up to 90 degrees, they are complementary.
Complementary Angles in Right Triangles
In a right-angled triangle, the two acute angles are always complementary. The right angle is 90 degrees, and the sum of all angles in a triangle is 180 degrees. So the two remaining angles must add up to 180 - 90 = 90 degrees.
Self-Complementary Angle
An angle that is the complement of itself. If A + A = 90 degrees, then 2A = 90 degrees, so A = 45 degrees. The angle of 45 degrees is the only self-complementary angle.
Quick Reference - Common Complementary Pairs:
| Angle | Its Complement |
|---|---|
| 10 degrees | 80 degrees |
| 20 degrees | 70 degrees |
| 25 degrees | 65 degrees |
| 30 degrees | 60 degrees |
| 35 degrees | 55 degrees |
| 40 degrees | 50 degrees |
| 45 degrees | 45 degrees |
Solved Examples
Example 1: Find the Complement of an Angle
Problem: Find the complement of 35 degrees.
Solution:
Given:
- Angle = 35 degrees
Using the formula:
- Complement = 90 - 35 = 55 degrees
Answer: The complement of 35 degrees is 55 degrees.
Example 2: Check if Two Angles are Complementary
Problem: Are 27 degrees and 63 degrees complementary?
Solution:
Given:
- Angle 1 = 27 degrees
- Angle 2 = 63 degrees
Check:
- Sum = 27 + 63 = 90 degrees
- Since the sum is 90 degrees, they are complementary.
Answer: Yes, 27 degrees and 63 degrees are complementary angles.
Example 3: Algebraic Problem - Find Both Angles
Problem: Two complementary angles are in the ratio 2:3. Find the angles.
Solution:
Given:
- Ratio of angles = 2:3
- Sum = 90 degrees
Steps:
- Let the angles be 2x and 3x.
- 2x + 3x = 90
- 5x = 90
- x = 18
Finding the angles:
- Angle 1 = 2 x 18 = 36 degrees
- Angle 2 = 3 x 18 = 54 degrees
Verification: 36 + 54 = 90 degrees. Correct!
Answer: The angles are 36 degrees and 54 degrees.
Example 4: Finding Angle Using Variable
Problem: Two complementary angles are (x + 10) degrees and (2x + 20) degrees. Find x and the two angles.
Solution:
Given:
- Angle 1 = (x + 10) degrees
- Angle 2 = (2x + 20) degrees
- Sum = 90 degrees
Setting up the equation:
- (x + 10) + (2x + 20) = 90
- 3x + 30 = 90
- 3x = 90 - 30 = 60
- x = 20
Finding the angles:
- Angle 1 = 20 + 10 = 30 degrees
- Angle 2 = 2(20) + 20 = 60 degrees
Verification: 30 + 60 = 90 degrees. Correct!
Answer: x = 20, the angles are 30 degrees and 60 degrees.
Example 5: Complement of Complement
Problem: Find the complement of the complement of 50 degrees.
Solution:
Step 1: Find the complement of 50 degrees.
- Complement = 90 - 50 = 40 degrees
Step 2: Find the complement of 40 degrees.
- Complement = 90 - 40 = 50 degrees
Answer: The complement of the complement of 50 degrees is 50 degrees (the original angle itself).
Note: The complement of the complement of any angle always gives back the original angle.
Example 6: Right Triangle Problem
Problem: In a right-angled triangle, one acute angle is 37 degrees. Find the other acute angle.
Solution:
Given:
- One angle = 90 degrees (right angle)
- Another angle = 37 degrees
In a triangle, all angles add up to 180 degrees:
- 90 + 37 + third angle = 180
- Third angle = 180 - 90 - 37 = 53 degrees
Note: The two acute angles (37 and 53) are complementary because 37 + 53 = 90.
Answer: The other acute angle is 53 degrees.
Example 7: Difference of Complementary Angles
Problem: Two complementary angles differ by 24 degrees. Find the angles.
Solution:
Given:
- Let the smaller angle = x degrees
- Larger angle = x + 24 degrees
- Sum = 90 degrees
Setting up the equation:
- x + (x + 24) = 90
- 2x + 24 = 90
- 2x = 66
- x = 33
Finding the angles:
- Smaller angle = 33 degrees
- Larger angle = 33 + 24 = 57 degrees
Verification: 33 + 57 = 90. Difference = 57 - 33 = 24. Both correct!
Answer: The angles are 33 degrees and 57 degrees.
Example 8: Clock Angle Problem
Problem: At 3 o'clock, the minute hand points at 12 and the hour hand points at 3, making a 90-degree angle. If the angle between the minute hand and a wall clock's decoration at 1 is 60 degrees, what is the angle between the decoration and the hour hand (at 3)?
Solution:
Given:
- Total angle from 12 to 3 = 90 degrees
- Angle from 12 to the decoration at 1 = 60 degrees
Finding the remaining angle:
- Angle from decoration to 3 = 90 - 60 = 30 degrees
These two angles (60 and 30) are complementary.
Answer: The angle between the decoration and the hour hand is 30 degrees.
Example 9: One Angle is Twice the Other
Problem: One of two complementary angles is twice the other. Find both angles.
Solution:
Given:
- Let the smaller angle = x degrees
- Larger angle = 2x degrees
- Sum = 90 degrees
Setting up the equation:
- x + 2x = 90
- 3x = 90
- x = 30
Finding the angles:
- Smaller angle = 30 degrees
- Larger angle = 2 x 30 = 60 degrees
Verification: 30 + 60 = 90. Correct!
Answer: The angles are 30 degrees and 60 degrees.
Example 10: Can These Be Complementary?
Problem: Can two angles of 50 degrees and 50 degrees be complementary?
Solution:
Given:
- Angle 1 = 50 degrees
- Angle 2 = 50 degrees
Check:
- Sum = 50 + 50 = 100 degrees
- 100 is not equal to 90
Answer: No, 50 degrees and 50 degrees are NOT complementary because their sum is 100 degrees, not 90 degrees.
Real-World Applications
Complementary angles are used in many real-life situations:
Architecture: Builders and architects use complementary angles when designing roofs, staircases, and ramps. If the angle of a ramp with the ground is 30 degrees, the angle with the wall is 60 degrees (complementary).
Sports: In cricket or basketball, the angle at which a ball bounces off a surface involves complementary angles. The angle of incidence and the angle with the surface are complementary.
Navigation: Pilots and sailors use complementary angles when calculating direction. If a ship turns 25 degrees from north, the angle remaining to reach east (90 degrees from north) is 65 degrees.
Carpentry: When cutting wood at angles, carpenters often need to find the complement of an angle to ensure pieces fit together at a right angle.
Art and Design: Artists use complementary angles to create balanced geometric patterns and designs.
Right Triangles: In any right-angled triangle, the two non-right angles are always complementary. This fact is used extensively in trigonometry.
Key Points to Remember
- Two angles are complementary if their sum is 90 degrees.
- Each angle is called the complement of the other.
- Complement of angle A = 90 - A degrees.
- Both angles in a complementary pair must be acute angles (less than 90 degrees).
- Complementary angles do NOT need to be adjacent (next to each other).
- In a right-angled triangle, the two acute angles are always complementary.
- 45 degrees is the only angle that is its own complement (45 + 45 = 90).
- The complement of the complement of an angle gives back the original angle.
- Do NOT confuse complementary (sum = 90 degrees) with supplementary (sum = 180 degrees).
Practice Problems
- Find the complement of 42 degrees.
- Find the complement of 73 degrees.
- Are 38 degrees and 52 degrees complementary? Why or why not?
- Two complementary angles are in the ratio 1:4. Find the angles.
- Two complementary angles differ by 36 degrees. Find the angles.
- If one angle is (3x - 5) degrees and its complement is (2x + 10) degrees, find x and both angles.
- In a right triangle, one angle is 28 degrees. Find the other acute angle.
- Can 95 degrees have a complement? Explain why or why not.
Frequently Asked Questions
Q1. What are complementary angles?
Two angles are complementary if the sum of their measures is 90 degrees. For example, 40 degrees and 50 degrees are complementary because 40 + 50 = 90.
Q2. What is the complement of 60 degrees?
The complement of 60 degrees is 90 - 60 = 30 degrees.
Q3. Can an obtuse angle have a complement?
No. An obtuse angle is greater than 90 degrees. Since complementary angles must add up to 90 degrees, the complement would have to be negative, which is not possible. Only acute angles (less than 90 degrees) have complements.
Q4. Do complementary angles need to be adjacent?
No. Two angles can be complementary even if they are not next to each other. The only requirement is that their sum equals 90 degrees.
Q5. What is the difference between complementary and supplementary angles?
Complementary angles add up to 90 degrees. Supplementary angles add up to 180 degrees. For example, 30 and 60 are complementary (sum = 90), while 30 and 150 are supplementary (sum = 180).
Q6. Can two right angles be complementary?
No. Each right angle is 90 degrees, so the sum would be 180 degrees, not 90 degrees. Two right angles are supplementary, not complementary.
Q7. Is 45 degrees special for complementary angles?
Yes. 45 degrees is the only angle that is its own complement, because 45 + 45 = 90. It is called a self-complementary angle.
Q8. Are the acute angles of a right triangle always complementary?
Yes. In a right-angled triangle, the right angle is 90 degrees. The sum of all angles is 180 degrees. So the two acute angles must add up to 180 - 90 = 90 degrees, which means they are complementary.
Related Topics
- Supplementary Angles
- Adjacent Angles
- Linear Pair of Angles
- Types of Angles
- Vertically Opposite Angles
- Transversal and Parallel Lines
- Corresponding Angles
- Alternate Interior Angles
- Co-Interior Angles
- Angles on a Straight Line
- Angles at a Point
- Alternate Exterior Angles
- Proving Lines are Parallel
- Word Problems on Lines and Angles
