Complementary Angles — Definition, Properties, Examples
Complementary Angles — Definition, Properties, Examples
TL;DR
Complementary angles are any two angles whose measures sum to exactly 90°. The two angles can sit side by side (forming a right angle — a corner) or be drawn anywhere on the page — what matters is the sum. This article covers the definition, properties, the two types (adjacent vs non-adjacent), the right-triangle connection, three worked examples.
What Are Complementary Angles?
Two angles are complementary when their measures add to 90°:
∠A + ∠B = 90°
When two complementary angles share a side and a vertex, they form a right angle — and together they look like the corner of a square or a sheet of paper. When they don't share anything, they're still complementary as long as their measures sum to 90° — they just don't look connected on the page.
For example:
- 30° and 60° are complementary (since 30° + 60° = 90°).
- 45° and 45° are complementary (each half of a right angle).
- 89° and 1° are complementary.
- 50° alone is not complementary to anything until you name a second angle.
The Four Properties of Complementary Angles
- Their sum is always exactly 90°. This is the defining property. If the sum is anything else, the angles are not complementary.
- They can be adjacent or non-adjacent. Adjacent complementary angles share a vertex and one side (they fit inside a right angle). Non-adjacent ones are simply two separate angles whose measures add to 90°.
- Both angles must be acute. Because the two have to sum to exactly 90° and neither can be zero, each angle must be strictly less than 90°. A right angle, an obtuse angle, or a reflex angle cannot be part of a complementary pair.
- Congruent Complements Theorem. If two angles are each complementary to the same third angle, then they are congruent to each other.
The Two Types of Complementary Angles
Type 1 — Adjacent Complementary Angles
When two complementary angles share a vertex and one side, they sit next to each other and their outer rays form a right angle.
Real-world example: the angle a staircase tread makes with vertical, plus the angle it makes with horizontal.
Type 2 — Non-Adjacent Complementary Angles
Two angles drawn anywhere — different corners of a figure, different problems on a worksheet — are complementary as long as their measures sum to 90°.
How to Find the Complement of an Angle
Subtract the given angle from 90°.
Complement of ∠A = 90° − ∠A
| Given angle | Complement |
|---|---|
| 10° | 80° |
| 25° | 65° |
| 30° | 60° |
| 45° | 45° |
| 60° | 30° |
| 72° | 18° |
| 89° | 1° |
Angles of exactly 90° or larger have no complement in standard geometry — the "complement" would be zero or negative. Complementary pairs exist only between angles each strictly between 0° and 90°.
The Right-Triangle Connection
The two non-right angles of a right triangle are always complementary.
Because the three interior angles of any triangle sum to 180°, and one is a 90° right angle, the other two must sum to 90°. By definition, they are complementary.
Three Worked Examples, From Quick to Stretch
Quick — Find the complement
Find the complement of ∠A = 27°.
90° − 27° = 63°
Answer: the complement is ∠B = 63°. Verify: 27° + 63° = 90° ✓.
Standard — Algebraic complement (Wrong Path Shown First)
Wrong path. A negative result indicates a problem.
Right path. Complementary means the sum equals 90°:
(3x+6) + (2x+4) = 90
Answer: x = 16; the two complementary angles are 54° and 36°.
Stretch — Two acute angles of a right triangle
The smaller angle is x°, then the larger is (2x + 15)°.
x + (2x + 15) = 90
Answer: the two acute angles are 25° and 65°.
Where Complementary Angles Show Up
Complementary angles are visible everywhere a right angle exists:
- Right triangles.
- Staircases.
- A clock at 3:00 or 9:00.
- Trigonometric co-function identities.
- A folded sheet of paper.
- Roof corner trim.
The Mistakes Students Make Most Often
Mistake 1: Confusing complementary with supplementary.
The fix: memory anchor — C for Complementary and C for Corner. S for Supplementary and S for Straight line.
Mistake 2: Assuming complementary angles must be adjacent.
The fix: the only rule is sum equals 90°. Location doesn't matter.
Mistake 3: Setting two complementary angles equal to each other.
The fix: complementary means the sum is 90°. Write the sum equation.
Mistake 4: Calling a single 90° angle "self-complementary".
The fix: a single 90° angle has no complement. Complementary pairs must be strictly between 0° and 90°.
Key Takeaways
- Complementary angles are two angles whose measures sum to exactly 90°.
- They can be adjacent or non-adjacent.
- Both angles must be acute.
- The two non-right angles of any right triangle are always complementary.
- Real-world places: right triangles, staircases, clock 3:00, trig identities.