What You Will Learn
In this lesson you will understand the definitions of complementary and supplementary angle pairs, learn how to find a missing angle when you know its complement or supplement, and explore related ideas including vertical angles and linear pairs.
Theory
Complementary Angles
Two angles are complementary when their measures add up to exactly 90°.
[ α + β = 90° ]
Each angle is called the complement of the other. For example, 35° and 55° are complementary because 35° + 55° = 90°.
Key fact: Complementary angles do not need to be next to each other. Any two angles whose measures sum to 90° are complementary, whether they share a side or are in completely different diagrams.
A real-world example: if a ramp rises at a 25° angle from the ground, the angle between the ramp and a vertical wall is 90° − 25° = 65°. These two angles are complementary.
Supplementary Angles
Two angles are supplementary when their measures add up to exactly 180°.
[ α + β = 180° ]
Each angle is the supplement of the other. For example, 110° and 70° are supplementary because 110° + 70° = 180°.
Memory trick: C omplementary = C orner (90°); S upplementary = S traight line (180°).
Linear Pairs
A linear pair is a specific case of supplementary angles: two adjacent angles that together form a straight line. Because a straight angle is 180°, the two angles in a linear pair are always supplementary.
If one angle in a linear pair is 130°, its partner is:
[ 180° − 130° = 50° ]
Vertical Angles
When two straight lines cross, they create four angles. The angles that sit directly across from each other (sharing only the vertex, not a side) are called vertical angles. Vertical angles are always equal.
If two lines intersect and one angle is 72°, then:
- The vertical angle across from it is also 72°.
- Each of the other two angles is [ 180° − 72° = 108° ] (since they form linear pairs with the 72° angle).
- Those two 108° angles are also vertical angles to each other.
Worked Examples
Example 1: Finding a complement
Find the complement of 57°.
[ 90° − 57° = 33° ]
The complement is 33°. Check: 57° + 33° = 90°. Correct.
Example 2: Finding a supplement
Find the supplement of 143°.
[ 180° − 143° = 37° ]
The supplement is 37°. Check: 143° + 37° = 180°. Correct.
Example 3: Algebraic — complementary angles
Two complementary angles are in the ratio 2:3. Find both angles.
Let the angles be 2x and 3x.
[ 2x + 3x = 90° ]
[ 5x = 90° ]
[ x = 18° ]
The angles are 2(18°) = 36° and 3(18°) = 54°.
Check: 36° + 54° = 90°. Correct.
Example 4: Vertical angles and linear pairs combined
Two lines intersect. One of the four angles measures (3x+10)° and the angle adjacent to it measures (2x+20)°. Find x and all four angle measures.
Adjacent angles at an intersection form a linear pair, so they are supplementary:
[ (3x+10) + (2x+20) = 180 ]
[ 5x + 30 = 180 ]
[ 5x = 150 ]
[ x = 30 ]
The first angle is [ 3(30) + 10 = 100° ]. The adjacent angle is [ 2(30) + 20 = 80° ].
By vertical angles: the four angles are 100°, 80°, 100°, 80°.
Check: 100° + 80° = 180°. Correct.
Common Mistakes
Mistake 1: Mixing up 90 and 180
❌ "The supplement of 40° is 90° − 40° = 50°."
✅ Supplement means the angles sum to 180°, not 90°. The supplement of 40° is [ 180° − 40° = 140° ]. The number 50° would be the complement.
Mistake 2: Assuming complementary angles must be adjacent
❌ "These two angles are not next to each other, so they cannot be complementary."
✅ Complementary (and supplementary) angles only need their measures to add up to the target sum. They do not need to share a side or even be in the same figure.
Mistake 3: Confusing vertical angles with adjacent angles
❌ "Vertical angles are the ones right next to each other."
✅ Vertical angles are across from each other at an intersection point (they share only the vertex). The angles next to each other are adjacent angles and form linear pairs.
Practice Problems
1. Find the complement of 76°.
Show Answer
[ 90°−76°=14° ]. The complement is 14°.
2. Find the supplement of 25°.
Show Answer
[ 180°−25°=155° ]. The supplement is 155°.
3. Angle A and angle B are supplementary. If A=(4x−10)° and B=(2x+40)°, find both angles.
Show Answer
[ (4x−10)+(2x+40)=180 ]
[ 6x+30=180 ]
[ 6x=150 ]
[ x=25 ]
Angle A=4(25)−10=90° and Angle B=2(25)+40=90°. Both angles are 90°.
4. Two lines intersect. One angle is 65°. Find the other three angles.
Show Answer
The vertical angle is also 65°. Each of the two remaining angles is [ 180°−65°=115° ].
5. The complement of an angle is three times the angle itself. Find the angle.
Show Answer
Let the angle be x. Its complement is 3x.
[ x + 3x = 90° ]
[ 4x = 90° ]
[ x = 22.5° ]
The angle is 22.5° and its complement is 67.5°. Check: 22.5° + 67.5° = 90°. Correct.
Summary
- Complementary angles sum to 90°; supplementary angles sum to 180°.
- A linear pair is two adjacent angles forming a straight line — they are always supplementary.
- Vertical angles are the opposite angles formed when two lines cross — they are always equal.
- To find a missing complement, subtract from 90°. To find a missing supplement, subtract from 180°.
- These relationships are the foundation for solving more advanced geometry problems involving parallel lines, triangles, and polygons.