Complementary and Supplementary Angles — Definitions, Examples, and Practice

By MathPal Editorial Team·Published Apr 17, 2026

Grade: 7-8 | Topic: Geometry

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 $90°$.

$\alpha + \beta = 90°$

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 $180°$.

$\alpha + \beta = 180°$

Each angle is the supplement of the other. For example, $110°$ and $70°$ are supplementary because $110° + 70° = 180°$.

Memory trick: Complementary = Corner ($90°$); Supplementary = Straight 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 $A = 4(25) - 10 = 90°$. Angle $B = 2(25) + 40 = 90°$.

Both angles are $90°$, and $90° + 90° = 180°$. Correct.

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 $180° - 65° = 115°$.

The four angles are: $65°$, $115°$, $65°$, $115°$.

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.

Related Topics

• Types of Angles — review how to classify angles by their measure.
• Angles in a Triangle — the triangle angle sum is closely related to supplementary angles.
• Area and Perimeter — right angles (complementary pairs) appear throughout area calculations.

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