Complementary and Supplementary Angles

Learn about angle pairs that add up to 90 degrees and 180 degrees.

Intermediate25 minLesson
Printable worksheet

Definition

Complementary angles are two angles whose measures add up to 90°90°.
Supplementary angles are two angles whose measures add up to 180°180°.
Complementary: ∠A+∠B=90°\text{Complementary: } \angle A + \angle B = 90°
Supplementary: ∠A+∠B=180°\text{Supplementary: } \angle A + \angle B = 180°
These angle pairs don't need to be adjacent (next to each other) - they just need to have the right sum!

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Two angles are complementary. What is their sum?

Worked Examples

If one angle measures 35°35°, what is its complement?
1

Recall the definition

Complementary angles add up to 90°90° → ∠A+∠B=90°\angle A + \angle B = 90°
2

Substitute the known angle

35°+∠B=90°35° + \angle B = 90° → Set up the equation
3

Solve for the unknown

∠B=90°−35°\angle B = 90° - 35° → ∠B=55°\angle B = 55°

Common Mistakes

Confusing complementary (90 degrees) with supplementary (180 degrees)

Why it's wrong: The terms sound similar and students mix up which sum goes with which name.

Correct: Memory trick: 'C' comes before 'S' in the alphabet, and 9090 comes before 180180. Complementary = 90°, Supplementary = 180°.

Thinking complementary/supplementary angles must be adjacent

Why it's wrong: Often shown as adjacent angles in diagrams, leading to this misconception.

Correct: Any two angles that sum to 90° are complementary, and any two that sum to 180° are supplementary - regardless of their position.

Subtracting from the wrong total

Why it's wrong: Students sometimes subtract from 180° when finding complements, or from 90° for supplements.

Correct: Always check: complement → subtract from 90°; supplement → subtract from 180°.

Interactive Visual

Angle Explorer

35°55°
α = 35°β = 55°α + β = 35° + 55° = 90°

Complementary angles add up to 90°. They do not have to be next to each other.

Drag the blue point to change one angle. The other angle changes so the sum stays the same.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
Two angles are complementary. What is their sum?

Why It Matters

Complementary and supplementary angles appear everywhere in the real world:
  • Architecture: Roof angles and support beams often form complementary or supplementary pairs
  • Sports: A basketball player's shooting arm and the backboard form angle pairs
  • Design: Artists use these angle relationships to create balanced compositions
  • Navigation: Pilots and sailors use angle calculations for course corrections
Understanding these relationships helps you solve for unknown angles in many practical situations!

Real World Applications

Construction and Carpentry

Carpenters use complementary and supplementary angles when building frames, roofs, and furniture.

Example:

A roof rafter meets the horizontal at a 35°35° angle. The complementary angle (55°55°) determines the cut angle for the other piece.

1Try It Yourself

A carpenter needs to cut two pieces of wood that will meet at a corner. One piece must be cut at a 62°62° angle.

At what angle should the second piece be cut so they form a right angle corner?

Step 1: Write the mathematical expression

Find the complement: 90°−62°90° - 62°

Roads and Intersections

Road planners measure the angles where a side road joins a straight main road. The two angles on either side of the side road lie along a straight line, so they are supplementary.

Example:

If a side road meets a straight main road at a 70°70° angle, the angle on the other side is 180°−70°=110°180° - 70° = 110°.

2Try It Yourself

A side road joins a straight main road. On one side, the angle between the two roads is 72°72°.

What is the angle between the roads on the other side of the side road?

Step 1: Write the mathematical expression

The two angles form a straight angle: 180°−72°180° - 72°

Key Takeaways

  • 1Complementary angles add up to 90°90° (a right angle)
  • 2Supplementary angles add up to 180°180° (a straight angle)
  • 3To find a complement: subtract from 90°90°
  • 4To find a supplement: subtract from 180°180°
  • 5These angle pairs don't have to be adjacent to each other

Frequently Asked Questions

Not within the same pair: two angles cannot add up to both 90°90° and 180°180°. But one acute angle can have both a complement and a supplement. For example, 40°40° has the complement 50°50° and the supplement 140°140°.
Not within the same pair: two angles cannot add up to both 90°90° and 180°180°. But one acute angle can have both a complement and a supplement. For example, 40°40° has the complement 50°50° and the supplement 140°140°.
No. Since obtuse angles are greater than 90°, and complementary angles must add to exactly 90°, an obtuse angle cannot have a complement (you would need a negative angle).
A linear pair consists of two adjacent angles that form a straight line. Linear pairs are always supplementary (add up to 180°).

Glossary

Complementary angles
Two angles whose measures add up to 90°90°
Supplementary angles
Two angles whose measures add up to 180°180°
Linear pair
Two adjacent angles that form a straight line (always supplementary)
Adjacent angles
Two angles that share a common vertex and side
Right angle
An angle that measures exactly 90°90°
Straight angle
An angle that measures exactly 180°180°

Formula Card

Complementary Angles

∠A+∠B=90°\angle A + \angle B = 90°

Two angles that sum to a right angle

Finding a Complement

∠B=90°−∠A\angle B = 90° - \angle A

Subtract the known angle from 90 degrees

Supplementary Angles

∠A+∠B=180°\angle A + \angle B = 180°

Two angles that sum to a straight angle

Finding a Supplement

∠B=180°−∠A\angle B = 180° - \angle A

Subtract the known angle from 180 degrees

Today's challengeOne problem, three tries

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