#### 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

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

a + 50 = 90

a = 90-50 =40°

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

a + 57 = 90

a = 90-57 =433°

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

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

**Answer: ** 50 degrees

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

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

**Answer: ** 68 degrees

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

**Answer: **60 degrees

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

**Answer: ** 20 degrees

$$ m\angle A + m\angle B = 90 \\ x + m\angle B = 90 $$

Now, let's just solve for$$ \angle B $$

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