Complementary angles

Complementary angles are two angles whose measures add up to 90°. Each angle is called the complement of the other, so 30° and 60° are complementary, and so are 45° and 45°. Put side by side, two complementary angles make a right angle. To find the complement of an angle, subtract it from 90°.

What are complementary angles?

Two angles are complementary when their measures add up to exactly 90°, a right angle. The angles do not have to touch: what matters is the sum. A 25° angle in one corner of a page and a 65° angle in another are still complementary.

• Only pairs count. Three angles of 30° add up to 90°, but they are not complementary. The word describes exactly two angles.
• Both angles are acute. Each must be more than 0° and less than 90°, because the other has to make up the rest.
• The complement of x° is (90 − x)°.

Adjacent and non-adjacent complementary angles

Adjacent complementary angles share a vertex and a side, so together they form a right angle, like a corner split into two parts by a ray. Non-adjacent complementary angles are apart from each other but still add up to 90°. Both kinds are complementary.

How to find the complement of an angle

Subtract the angle from 90°.

Here x is an angle in degrees and y is its complement. As one angle grows, its complement shrinks by the same amount, so the line slopes down. At 45° the two are equal, and at 90° the complement is 0, which is why a right angle has no complement.

Example 1: whole degrees

Find the complement of 35°: 90° − 35° = 55°. Check: 35° + 55° = 90°.

Example 2: decimals

Find the complement of 72.5°: 90° − 72.5° = 17.5°.

Example 3: are these complementary?

• 48° and 42°: 48 + 42 = 90, so yes.
• 50° and 45°: 50 + 45 = 95, so no.

Complementary angles in radians

A right angle is π/2 radians, so in radians two angles are complementary when they add up to π/2. The complement of π/6 is π/2 − π/6 = π/3, which matches 30° and 60°.

Complementary vs supplementary angles

Complementary angles have a partner idea: supplementary angles, which add up to 180°.

Example 4: complement and supplement together

The supplement of an angle is 3 times its complement. Find the angle.

The angle is 45°. Check: its supplement is 135° and its complement is 45°, and 135 = 3 × 45.

Complementary angles problems with algebra

Most problems give the two angles as expressions. Set their sum equal to 90, solve for x, then put x back into each expression. Grade 7 math under the Common Core standards asks students to write and solve equations like these to find unknown angles.

Example 5: x and 2x

Two complementary angles measure x° and 2x°. Find both.

The angles are 30° and 60°.

Example 6: two expressions

Two complementary angles measure (3x + 5)° and (2x − 5)°. Find both.

The angles are 3(18) + 5 = 59° and 2(18) − 5 = 31°. Check: 59 + 31 = 90.

Example 7: a word problem

An angle is 12° more than its complement. Find the angle.

Call the complement x. Then the angle is x + 12, and together they make 90°:

The complement is 39°, so the angle is 51°.

Example 8: a ratio

Two complementary angles are in the ratio 1 : 2. The ratio has 1 + 2 = 3 parts, and 90° ÷ 3 = 30° per part, so the angles are 30° and 60°. This method works for any ratio.

Try one yourself.

Complementary angles in a right triangle

The angles of any triangle add up to 180°. A right triangle uses 90° of that on the right angle, which leaves 90° for the other two. So the two acute angles of a right triangle are always complementary.

If one acute angle of a right triangle is 28°, the other is 90° − 28° = 62°. You don't need any side lengths.

Sine and cosine of complementary angles

In a right triangle, the side opposite one acute angle is the side adjacent to the other. That is why the sine of an angle equals the cosine of its complement:

For example, sin 30° = cos 60° = 0.5, and sin 20° = cos 70° ≈ 0.342. The "co" in cosine is short for complement: cosine means "sine of the complement". The high school Common Core standards ask students to explain and use this relationship, and the Trigonometry course builds it up from right triangles and the unit circle.

Complementary angles in real life

• Corners: a ray from the corner of a picture frame, a window or a tile splits the right angle into two complementary angles.
• Ramps and ladders: a ladder against a wall makes an angle with the ground and an angle with the wall that add up to 90°, because the wall meets the ground at a right angle.
• The hands of a clock at 3:00 make a right angle. A line through the center between them splits it into two complementary angles.

Common mistakes

• Mixing up complementary and supplementary. Complementary is 90°, supplementary is 180°.
• Thinking the angles must touch. Non-adjacent angles can be complementary too.
• Calling three angles complementary. The word is only for pairs.
• Stopping at x. In algebra problems, x is not the answer. Substitute it back to find each angle.
• Looking for the complement of an obtuse angle. An angle of 90° or more has no complement.

Practice problems

1. Find the complement of 18°.
2. Find the complement of 64.3°.
3. Two complementary angles are (4x)° and (x + 15)°. Find both angles.
4. One acute angle of a right triangle is 47°. Find the other.
5. An angle is 4 times its complement. Find the angle.

Answers: 1) 72°. 2) 25.7°. 3) x = 15, so 60° and 30°. 4) 43°. 5) 72°, since x = 4(90 − x) gives 5x = 360.

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