Mathwords logoReference LibraryMathwords

Complement of an Angle

Complement of an Angle

The complement of an acute angle A is the angle 90° – A. For example, the complement of 20° is 70°.

 

 

See also

Complementary angles, supplement of an angle, measure of an angle

Key Formula

Complement of A=90∘−A\text{Complement of } A = 90^\circ - A
Where:
  • AA = The measure of the original acute angle (must satisfy 0° < A < 90°)

Worked Example

Problem: Find the complement of a 35° angle.
Step 1: Write the complement formula.
Complement=90∘−A\text{Complement} = 90^\circ - A
Step 2: Substitute A = 35°.
Complement=90∘−35∘\text{Complement} = 90^\circ - 35^\circ
Step 3: Compute the result.
Complement=55∘\text{Complement} = 55^\circ
Answer: The complement of 35° is 55°. Notice that 35° + 55° = 90°, confirming the two angles are complementary.

Another Example

Problem: Two complementary angles are such that one angle is twice the other. Find both angles.
Step 1: Let the smaller angle be x. The larger angle is 2x.
x+2x=90∘x + 2x = 90^\circ
Step 2: Combine like terms.
3x=90∘3x = 90^\circ
Step 3: Solve for x.
x=30∘x = 30^\circ
Step 4: Find the larger angle.
2x=60∘2x = 60^\circ
Answer: The two angles are 30° and 60°. Each is the complement of the other.

Frequently Asked Questions

Can an angle greater than 90° have a complement?
No. The complement formula gives 90° − A, which would be negative for any angle greater than or equal to 90°. Because angle measures must be positive, only acute angles (angles strictly between 0° and 90°) have complements.
What is the complement of 45°?
The complement of 45° is also 45°, since 90° − 45° = 45°. This means a 45° angle is its own complement — the only angle with this property.

Complement (90°) vs. Supplement (180°)

The complement of an angle is what you add to reach 90°, while the supplement of an angle is what you add to reach 180°. For example, the complement of 30° is 60°, but the supplement of 30° is 150°. A quick way to remember: "C" comes before "S" in the alphabet, and 90 comes before 180 on the number line.

Why It Matters

Complements appear constantly in right triangles: the two non-right angles always sum to 90°, so each is the complement of the other. This relationship is the basis of cofunctions in trigonometry — for instance, sin⁡(A)=cos⁡(90°−A)\sin(A) = \cos(90° - A), which is why cosine is literally the "complement's sine." Understanding complements also helps you solve geometry problems involving perpendicular lines and angle bisectors.

Common Mistakes

Mistake: Confusing complement with supplement — subtracting from 180° instead of 90°.
Correction: Complement always involves 90°. Remember: the 'co' in complement pairs with the 'co' in corner (a right angle). Supplement involves 180° (a straight line).
Mistake: Trying to find the complement of an obtuse angle (90° or greater).
Correction: Only acute angles have complements. If A ≥ 90°, then 90° − A ≤ 0°, which is not a valid angle measure.

Related Terms