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
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. 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 $90°$.
$$\text{Complement of } \angle A = 90° - \angle A$$
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.
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
Find the complement of $\angle X = 18°$.
Two complementary angles are given as $(5x - 10)°$ and $(2x + 8)°$. Find $x$ and both angles.
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.
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