Complementary Angles — Definition, Properties, Examples

What Are Complementary Angles?

Two angles are complementary when their measures add to 90°:

$$\angle A + \angle B = 90°$$

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

1. Their sum is always exactly $90°$. This is the defining property. If the sum is anything else, the angles are not complementary.

2. 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 $90°$.

3. 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.

4. Congruent Complements Theorem. If two angles are each complementary to the same third angle, then they are congruent to each other. In symbols: if $\angle A + \angle C = 90°$ and $\angle B + \angle C = 90°$, then $\angle A = \angle B$.

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 — the corner of a square, an open book, or a perfectly squared piece of paper.

Real-world example: the angle a staircase tread makes with vertical, plus the angle it makes with horizontal. The two together fill a right angle at the corner.

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 $90°$.

Common example: the two acute angles of a right triangle. They sit at different vertices and don't share a side — but they always sum to $90°$ because the third angle is the right angle, and the three interior angles of a triangle sum to $180°$.

How to Find the Complement of an Angle

Subtract the given angle from $90°$.

$$\text{Complement of } \angle A = 90° - \angle A$$

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.

Why: the three interior angles of any triangle sum to $180°$. In a right triangle, one of those angles is $90°$. The other two must therefore sum to $180° - 90° = 90°$. By definition, they are complementary.

This is why the trigonometric co-function identities carry the prefix "co-":

$$\sin(90° - \theta) = \cos(\theta)$$ $$\cos(90° - \theta) = \sin(\theta)$$ $$\tan(90° - \theta) = \cot(\theta)$$

The "co-" in cosine, cotangent, and cosecant literally means "of the complementary angle". The relationship between sine and cosine is exactly the relationship between the two acute angles of a right triangle — they're complements of each other.

Three Worked Examples, From Quick to Stretch

Quick — Find the complement

Find the complement of $\angle A = 27°$.

$$90° - 27° = 63°$$

Answer: the complement is $\angle B = 63°$. Verify: $27° + 63° = 90°$ ✓.

Standard — Algebraic complement (Wrong Path Shown First)

Two complementary angles have measures $(3x + 6)°$ and $(2x + 4)°$. Find $x$ and both angles.

Wrong path. A student in a hurry sets the two expressions equal — "because complementary angles must be equal in a complementary pair" — and gets $3x + 6 = 2x + 4$, which solves to $x = -2$. A negative $x$ should be a warning sign, and plugging back gives angles of $0°$ and $0°$ — definitely not a complementary pair.

Right path. Complementary means the sum equals $90°$, not that the angles are equal:

$$(3x + 6) + (2x + 4) = 90$$ $$5x + 10 = 90$$ $$5x = 80 \Rightarrow x = 16$$

So the first angle is $3(16) + 6 = 54°$ and the second is $2(16) + 4 = 36°$. Verify: $54° + 36° = 90°$ ✓.

Answer: $x = 16$; the two complementary angles are $54°$ and $36°$.

Stretch — Two acute angles of a right triangle

In a right triangle, one of the two non-right angles is $15°$ more than twice the other. Find both acute angles.

Let the smaller angle be $x°$. Then the larger is $(2x + 15)°$. Since the two are complementary (the two non-right angles in a right triangle always are):

$$x + (2x + 15) = 90$$ $$3x + 15 = 90$$ $$3x = 75 \Rightarrow x = 25$$

So the smaller is $25°$ and the larger is $2(25) + 15 = 65°$. Verify: $25° + 65° = 90°$ ✓, and $25° + 65° + 90° = 180°$ (the full triangle) ✓.

Answer: the two acute angles are $25°$ and $65°$.

Where Complementary Angles Show Up

Complementary angles are visible everywhere a right angle exists — and right angles are everywhere.

• Right triangles. The two non-right angles. The most-used complementary pair in geometry.

• Staircases. The angle of the tread (horizontal) and the angle of the riser (vertical) at a corner are complementary because they together fill a right angle.

• A clock at 3:00 or 9:00. The two small angles formed between the hands and the 12 are complementary at exactly 3 o'clock (and at 9 o'clock by symmetry).

• Trigonometric co-function identities. Sine and cosine, tangent and cotangent, secant and cosecant — each pair is the trig value of complementary angles. The historical reason cosine is named cosine is exactly this.

• A folded sheet of paper. When you fold a square sheet diagonally, the two acute angles at the fold are complementary (each $45°$).

• Roof corner trim. The angle a roof slope makes with vertical and the angle it makes with horizontal are complementary at the corner where they meet.

• A surveyor's theodolite. The instrument measures altitude and zenith angles — and altitude + zenith = $90°$. Complementary by design.

The Mistakes Students Make Most Often

Mistake 1: Confusing complementary with supplementary.

Where it slips in: the two terms sound similar. Students mix them up under exam pressure.

The fix: memory anchor — C for Complementary and C for Corner ($90°$). S for Supplementary and S for Straight line ($180°$). Letter-to-shape mnemonic that sticks.

Mistake 2: Assuming complementary angles must be adjacent.

Where it slips in: a problem gives two angles in different parts of a figure with measures $40°$ and $50°$, and the student dismisses them as "not complementary" because they aren't next to each other.

The fix: the only rule is sum equals $90°$. Location doesn't matter. Adjacent complementary angles together form a right angle; non-adjacent complementary angles still satisfy the sum but don't visibly form a right angle.

Mistake 3: Setting two complementary angles equal to each other.

Where it slips in: in algebraic problems where two angles are given as expressions in $x$. Students set the expressions equal instead of setting their sum to $90°$.

The fix: complementary means the sum is $90°$. Write the sum equation, not the equality. (See the Standard worked example above.)

Mistake 4: Calling a single $90°$ angle "self-complementary".

Where it slips in: students reason that since $90° + 0° = 90°$, a right angle is complementary to a $0°$ angle.

The fix: a $0°$ angle isn't a real angle in the geometric sense, and a single $90°$ angle has no complement. Complementary pairs both have measures strictly between $0°$ and $90°$.

Key Takeaways

• Complementary angles are two angles whose measures sum to exactly $90°$.

• They can be adjacent (forming a right angle — a corner) or non-adjacent (anywhere on the page, as long as the sum is $90°$).

• Both angles must be acute — each strictly between $0°$ and $90°$.

• The two non-right angles of any right triangle are always complementary — this is the foundation of the trig co-function identities.

• Real-world places: right triangles, staircases, clock 3:00, trig identities, paper folding, surveying.

Practice These Three Before Moving On

1. Find the complement of $\angle X = 18°$.

2. Two complementary angles are given as $(5x - 10)°$ and $(2x + 8)°$. Find $x$ and both angles.

3. In a right triangle, one acute angle is twice the other. Find both acute angles.

If problem 3 returned $30°$ and $60°$ — you've got it. Want a Bhanzu trainer to walk through more angle-relationship problems? Book a free demo class — online globally.

Also Read:

• Geometry — Concepts, Formulas, Shapes & Topics Guide

• What is Adjacent?

• Adjacent Angles

• Vertical Angles

• Supplementary Angles

• Supplementary vs Complementary Angles

• Linear Pair of Angles