Supplementary Angles
Open a book and lay it flat on a table. The two pages form a straight line across the top. The angle along that straight line is 180 degrees. Now imagine placing a pencil on the open book so it points upward. The pencil divides the 180-degree angle into two parts. These two parts are called supplementary angles.
Supplementary angles are pairs of angles whose measures add up to 180 degrees. You see them whenever a straight line is divided into two angles by another line. They appear on straight roads, in door hinges, and in the hands of a clock at 6 o'clock.
The word "supplementary" comes from the Latin word "supplementum," meaning "to supply" or "to complete." Two supplementary angles complete a straight angle (180 degrees) together.
In this chapter, you will learn the definition of supplementary angles, how to find the supplement of a given angle, the difference between supplementary and complementary angles, and how to solve problems involving supplementary angles.
What is Supplementary Angles?
Definition: Two angles are called supplementary angles if the sum of their measures is 180 degrees.
Each angle is called the supplement of the other.
Angle 1 + Angle 2 = 180 degrees
Key Points about Supplementary Angles:
- The two angles do NOT need to be next to each other. They can be anywhere. If their sum is 180 degrees, they are supplementary.
- One angle can be acute and the other obtuse (this is the most common case).
- Both angles can be right angles (90 + 90 = 180).
- Neither angle can be greater than 180 degrees.
- The supplement of an angle A is (180 - A) degrees.
Examples of Supplementary Pairs:
| Angle 1 | Angle 2 | Sum | Supplementary? |
|---|---|---|---|
| 60 degrees | 120 degrees | 180 degrees | Yes |
| 90 degrees | 90 degrees | 180 degrees | Yes |
| 45 degrees | 135 degrees | 180 degrees | Yes |
| 110 degrees | 70 degrees | 180 degrees | Yes |
| 100 degrees | 100 degrees | 200 degrees | No |
| 30 degrees | 60 degrees | 90 degrees | No (these are complementary) |
Complementary vs Supplementary:
| Property | Complementary Angles | Supplementary Angles |
|---|---|---|
| Sum | 90 degrees | 180 degrees |
| Both angles are | Acute (< 90 degrees) | Can be acute, right, or obtuse |
| Formed along | Right angle | Straight line |
| Memory trick | "C" for Corner (90 degrees) | "S" for Straight (180 degrees) |
Supplementary Angles Formula
Formula to Find the Supplement of an Angle:
Supplement of angle A = 180 degrees - A
Where:
- A is the given angle
- A must be less than 180 degrees for the supplement to exist as a positive angle
Using Supplementary Angles in Equations:
If two supplementary angles are given in terms of a variable, set up the equation:
Angle 1 + Angle 2 = 180 degrees
Solve for the variable, then find each angle.
Angles on a Straight Line:
If two angles are on a straight line, they are supplementary (sum = 180 degrees)
Types and Properties
Supplementary angles can appear in different forms:
Adjacent Supplementary Angles (Linear Pair)
Two angles that are next to each other (share a common side and vertex) and together form a straight line. This is called a linear pair. When a line stands on another line, the two angles on either side are supplementary and form a linear pair.
Non-Adjacent Supplementary Angles
Two angles that are NOT next to each other but still add up to 180 degrees. For example, 65 degrees drawn on one page and 115 degrees drawn on another page. Their sum is 180, so they are supplementary.
Supplementary Angles in Parallel Lines
When a transversal crosses two parallel lines, the co-interior angles (also called same-side interior angles or allied angles) are always supplementary.
Self-Supplementary Angle
An angle that is its own supplement. If A + A = 180 degrees, then 2A = 180, so A = 90 degrees. A right angle (90 degrees) is the only self-supplementary angle.
Quick Reference - Common Supplementary Pairs:
| Angle | Its Supplement |
|---|---|
| 10 degrees | 170 degrees |
| 30 degrees | 150 degrees |
| 45 degrees | 135 degrees |
| 60 degrees | 120 degrees |
| 75 degrees | 105 degrees |
| 90 degrees | 90 degrees |
| 120 degrees | 60 degrees |
Solved Examples
Example 1: Find the Supplement of an Angle
Problem: Find the supplement of 65 degrees.
Solution:
Given:
- Angle = 65 degrees
Using the formula:
- Supplement = 180 - 65 = 115 degrees
Answer: The supplement of 65 degrees is 115 degrees.
Example 2: Check if Two Angles are Supplementary
Problem: Are 73 degrees and 107 degrees supplementary?
Solution:
Given:
- Angle 1 = 73 degrees
- Angle 2 = 107 degrees
Check:
- Sum = 73 + 107 = 180 degrees
- Since the sum is 180 degrees, they are supplementary.
Answer: Yes, 73 degrees and 107 degrees are supplementary angles.
Example 3: Ratio Problem
Problem: Two supplementary angles are in the ratio 2:7. Find the angles.
Solution:
Given:
- Ratio of angles = 2:7
- Sum = 180 degrees
Steps:
- Let the angles be 2x and 7x.
- 2x + 7x = 180
- 9x = 180
- x = 20
Finding the angles:
- Angle 1 = 2 x 20 = 40 degrees
- Angle 2 = 7 x 20 = 140 degrees
Verification: 40 + 140 = 180 degrees. Correct!
Answer: The angles are 40 degrees and 140 degrees.
Example 4: Variable Problem
Problem: Two supplementary angles are (3x + 15) degrees and (2x + 40) degrees. Find x and the two angles.
Solution:
Given:
- Angle 1 = (3x + 15) degrees
- Angle 2 = (2x + 40) degrees
- Sum = 180 degrees
Setting up the equation:
- (3x + 15) + (2x + 40) = 180
- 5x + 55 = 180
- 5x = 125
- x = 25
Finding the angles:
- Angle 1 = 3(25) + 15 = 75 + 15 = 90 degrees
- Angle 2 = 2(25) + 40 = 50 + 40 = 90 degrees
Verification: 90 + 90 = 180. Correct!
Answer: x = 25, the angles are 90 degrees and 90 degrees.
Example 5: Supplement of Supplement
Problem: Find the supplement of the supplement of 130 degrees.
Solution:
Step 1: Find the supplement of 130 degrees.
- Supplement = 180 - 130 = 50 degrees
Step 2: Find the supplement of 50 degrees.
- Supplement = 180 - 50 = 130 degrees
Answer: The supplement of the supplement of 130 degrees is 130 degrees (the original angle).
Note: The supplement of the supplement of any angle always gives back the original angle.
Example 6: Straight Line Problem
Problem: A straight line is divided into two angles by a ray. One angle is 72 degrees. Find the other angle.
Solution:
Given:
- The angles on a straight line are supplementary.
- One angle = 72 degrees
Finding the other angle:
- Other angle = 180 - 72 = 108 degrees
Answer: The other angle is 108 degrees.
Example 7: Difference of Supplementary Angles
Problem: Two supplementary angles differ by 48 degrees. Find the angles.
Solution:
Given:
- Let the smaller angle = x degrees
- Larger angle = (x + 48) degrees
- Sum = 180 degrees
Setting up the equation:
- x + (x + 48) = 180
- 2x + 48 = 180
- 2x = 132
- x = 66
Finding the angles:
- Smaller angle = 66 degrees
- Larger angle = 66 + 48 = 114 degrees
Verification: 66 + 114 = 180. Difference = 114 - 66 = 48. Both correct!
Answer: The angles are 66 degrees and 114 degrees.
Example 8: One Angle is Three Times the Other
Problem: One of two supplementary angles is three times the other. Find both angles.
Solution:
Given:
- Let the smaller angle = x degrees
- Larger angle = 3x degrees
- Sum = 180 degrees
Setting up the equation:
- x + 3x = 180
- 4x = 180
- x = 45
Finding the angles:
- Smaller angle = 45 degrees
- Larger angle = 3 x 45 = 135 degrees
Verification: 45 + 135 = 180. Correct!
Answer: The angles are 45 degrees and 135 degrees.
Example 9: Can an Angle be Both Complement and Supplement?
Problem: Is there an angle whose supplement is three times its complement? Find the angle.
Solution:
Given:
- Let the angle = x degrees
- Complement = (90 - x) degrees
- Supplement = (180 - x) degrees
- Supplement = 3 x Complement
Setting up the equation:
- 180 - x = 3(90 - x)
- 180 - x = 270 - 3x
- -x + 3x = 270 - 180
- 2x = 90
- x = 45
Verification:
- Complement of 45 = 45 degrees
- Supplement of 45 = 135 degrees
- Is 135 = 3 x 45? Yes!
Answer: The angle is 45 degrees.
Example 10: Door Hinge Problem
Problem: A door is opened at an angle of 55 degrees from the wall. What angle does it make with the wall on the other side?
Solution:
Given:
- When a door opens, the angle on one side and the angle on the other side of the wall together form a straight angle (180 degrees).
- Angle on one side = 55 degrees
Finding the other angle:
- Angle on other side = 180 - 55 = 125 degrees
Answer: The door makes an angle of 125 degrees with the wall on the other side.
Real-World Applications
Supplementary angles appear in many real-life situations:
Straight Roads and Intersections: When two roads meet, the angles on a straight line are supplementary. If one angle at a T-junction is 120 degrees, the other is 60 degrees.
Doors and Windows: The angle a door makes on one side of the wall and the angle on the other side are supplementary (they add up to 180 degrees).
Clocks: At 6 o'clock, the hands of a clock form a straight angle of 180 degrees. Any line drawn through the centre divides this into two supplementary angles.
Bridges and Architecture: Engineers use supplementary angles when designing supports and beams that lie along straight lines.
Parallel Lines: Co-interior angles formed when a transversal crosses parallel lines are always supplementary. This property is used in construction and road planning.
Folding Paper: When you fold a rectangular paper, the angles on either side of the fold along the edge are supplementary.
Key Points to Remember
- Two angles are supplementary if their sum is 180 degrees.
- Each angle is the supplement of the other.
- Supplement of angle A = 180 - A degrees.
- Supplementary angles can be acute + obtuse, or right + right.
- Supplementary angles do NOT need to be adjacent.
- Adjacent supplementary angles form a linear pair (on a straight line).
- 90 degrees is the only self-supplementary angle (90 + 90 = 180).
- The supplement of the supplement of an angle is the original angle.
- Complementary = sum is 90 degrees. Supplementary = sum is 180 degrees. Do not confuse them.
- Memory trick: C for Corner (90 degrees), S for Straight line (180 degrees).
Practice Problems
- Find the supplement of 78 degrees.
- Find the supplement of 145 degrees.
- Are 89 degrees and 91 degrees supplementary?
- Two supplementary angles are in the ratio 3:5. Find the angles.
- Two supplementary angles differ by 54 degrees. Find the angles.
- If one angle is (4x + 10) degrees and its supplement is (6x - 30) degrees, find x and both angles.
- Can two acute angles be supplementary? Explain.
- Find an angle whose supplement is 4 times its complement.
Frequently Asked Questions
Q1. What are supplementary angles?
Two angles are supplementary if the sum of their measures is 180 degrees. For example, 110 degrees and 70 degrees are supplementary because 110 + 70 = 180.
Q2. What is the supplement of 120 degrees?
The supplement of 120 degrees is 180 - 120 = 60 degrees.
Q3. Can two acute angles be supplementary?
No. Each acute angle is less than 90 degrees, so the maximum sum of two acute angles is less than 180 degrees. At least one of the angles must be 90 degrees or more.
Q4. Can two obtuse angles be supplementary?
No. Each obtuse angle is more than 90 degrees, so the minimum sum of two obtuse angles is more than 180 degrees. This exceeds 180 degrees.
Q5. What is the difference between supplementary and complementary angles?
Complementary angles add up to 90 degrees. Supplementary angles add up to 180 degrees. A useful trick: C (Complementary) comes before S (Supplementary) in the alphabet, and 90 comes before 180.
Q6. What is a linear pair?
A linear pair is a pair of adjacent supplementary angles formed when a ray stands on a straight line. The two angles share a common vertex and a common side, and their non-common sides form a straight line. Every linear pair is supplementary, but not every supplementary pair is a linear pair.
Q7. Is 90 degrees its own supplement?
Yes. 90 + 90 = 180, so a right angle is supplementary to another right angle. It is the only self-supplementary angle.
Q8. Can supplementary angles be non-adjacent?
Yes. Two angles can be supplementary without being next to each other. For example, 40 degrees in one triangle and 140 degrees in another shape are supplementary because their sum is 180 degrees.
Related Topics
- Complementary Angles
- Linear Pair of Angles
- Adjacent 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
