Complementary Angles

Formula, Examples, diagrams and practice problems

What are complementary angles?

They are angles whose sum is 90°.

Do Complementary angles need to be next to each other (ie adjacent)?

No!

Complementary angles do not need to be adjacent angles (angles next to one another).

All of the pairs of angles pictured below are complementary.

Practice Problems

Problem 1

Since the angles are complementary (note: the perpendicular symbol).

$$a + 50° = 90° \\ a = 90° -50° \\ a = 40°$$

Problem 2

Since the angles are complementary (note: the perpendicular symbol).

$$a + 57° = 90° \\ a = 90° - 57° \\ a = 33°$$

Problem 3

Since these angles are complementary we can set up the following equation.

$$m\angle A + m\angle B = 90^{\circ} \\$$

Now, substitute the known angle into equation and solve.

$$40^{\circ} + m\angle B = 90^{\circ} \\ 40^{\circ} \color{red}{- 40^{\circ}}+ m\angle B = 90^{\circ} \color{red}{- 40^{\circ}} \\ m\angle B = \color{red}{ 50^{\circ}} \\$$

Problem 4

Since these angles are complementary we can set up the following equation:

$$m\angle X + m\angle Z = 90^{\circ} \\$$

Now, substitute the known angle into equation and solve.

$$22^{\circ} + m\angle X = 90^{\circ} \\ 22^{\circ} \color{red}{- 22^{\circ}}+ m\angle B = 90^{\circ} \color{red}{- 22^{\circ}} \\ m\angle B = \color{red}{ 68^{\circ}} \\$$

Problem 5

First, since this is a ratio problem, we know that 2x + 1x = 90, so now, let's first solve for x:

$$3x = 90 \\ x = \frac{90}{3} = 30$$

Now, the larger angle is the 2x which is 2(30) = 60 degrees.

Problem 6

First, since this is a ratio problem, we know that 7x + 2x = 90, so now, let's first solve for x:

$$9x = 90 \\ x = \frac{90}{9} = 10$$

Now, the smaller angle is the 2x which is 2(10) = 20 degrees.

$$m\angle A + m\angle B = 90 \\ x + m\angle B = 90$$
$$\angle B$$
$$x + m\angle B = 90 \\ x \color{red}{- x} + m\angle B = 90\color{red}{- x} \\ m\angle B = \color{red}{90 - x}$$