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Complementary angles

# Complementary angles

Here you will learn about complementary angles, including how to find missing angles by applying knowledge of complementary angles.

Students will first learn about complementary angles as a part of measurement and data in 4th grade. They will expand that knowledge as they progress through middle school.

## What are complementary angles?

Complementary angles are two angles that add up to 90 degrees. They can be adjacent or not adjacent. Adjacent angles are angles that share a common arm and a vertex.

Adjacent complementary angles make a right angle.

You can use what you know about complementary angles to decompose angles and find the measurement of an unknown angle.

## Common Core State Standards

How does this relate to 4th grade math?

• Grade 4: Measurement and Data (4.MD.C.7)
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts.

Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, for example, by using an equation with a symbol for the unknown angle measure.

## How to find the measure of complementary angles

In order to find the measure of complementary angles you need to:

1. Determine which angles are complementary.
2. Identify given angle measurements and the unknown angle.
3. Find the missing angle.

## Complementary angles examples

### Example 1: finding complementary angles (adjacent angles)

The two angles shown, x and y, are complementary. Find the measure of angle x.

1. Determine which angles are complementary.

The question states that angles x and y are complementary and equal 90^{\circ}.

x+y=90

2Identify given angle measurements and the unknown angle or angles.

Angle y is 35^{\circ}.

The unknown angle is angle x, and when it’s added to 35^{\circ} will equal 90^{\circ}.

x+35=90

3Find the missing angle.

To find the missing angle, you will subtract 35 from 90.

x=90-35

x =55

The measurement of angle x is 55^{\circ}.

### Example 2: finding complementary angles (adjacent angles)

The two angles shown are complementary. Find the measure of angle x.

Determine which angles are complementary.

Identify given angle measurements and the unknown angle or angles.

Find the missing angle.

### Example 3: finding complementary angles (not adjacent angles)

The two angles shown are complementary. Find the measure of angle a.

Determine which angles are complementary.

Identify given angle measurements and the unknown angle or angles.

Find the missing angle.

### Example 4: finding complementary angles (not adjacent angles)

The two angles shown are complementary. Find the measure of angle c.

Determine which angles are complementary.

Identify given angle measurements and the unknown angle or angles.

Find the missing angle.

### Example 5: finding an angle which is complementary to another

Two angles ‘x and y' are complementary and one of them is 49^{\circ}. What is the size of the other angle?

Determine which angles are complementary.

Identify given angle measurements and the unknown angle or angles.

Find the missing angle.

### Example 6: finding an angle which is complementary to another

Two angles ‘x and y' are complementary and one of them is 17^{\circ}. What is the size of the other angle?

Determine which angles are complementary.

Identify given angle measurements and the unknown angle or angles.

Find the missing angle.

### Teaching tips for complementary angles

• Rather than having students practice finding and decomposing complementary angles on multiple skill worksheets, provide them with a variety of practice problems, activities, and/or projects that have a real-world context. This will deepen their understanding of this skill.

### Easy mistakes to make

• Mixing up supplementary angles and complementary angles
Students may mix up supplementary and complementary angles, thinking that complementary angles add up to 180^{\circ} and that supplementary angles add up to 90^{\circ}. However, complementary angles add up to 90^{\circ} and supplementary angles add up to 180^{\circ}.

• Assuming complementary angles always have a common vertex
Complementary angles can be either adjacent or not adjacent. Not adjacent complementary angles do not share a common vertex, but their angles will still add up to 180^{\circ}.

### Complementary angles practice questions

1. \angle l and \angle m are complementary angles. \angle l measures at 19^{\circ}. What is the measure of \angle m?

m=61^{\circ}

m=161^{\circ}

m=154^{\circ}

m=71^{\circ}

The two angles are complementary, so they must have a sum of 90.

You will subtract 19 from 90 to find the measure of the missing angle.

m=90-19

m=71

The missing angle measures at 71^{\circ}.

2. \angle j and \angle k are complementary angles. \angle j measures at 45^{\circ}. What is the measure of \angle k?

k=43^{\circ}

k=45^{\circ}

k=145^{\circ}

k=143^{\circ}

The two angles are complementary, so they must have a sum of 90.

You will subtract 45 from 90 to find the measure of the missing angle.

k=90-45

k=45

The missing angle measures at 45^{\circ}.

3. \angle w and \angle z are complementary angles. \angle z measures at 36^{\circ}. What is the measure of \angle w?

w=154^{\circ}

w=47^{\circ}

w=54^{\circ}

w=132^{\circ}

The two angles are complementary, so they must have a sum of 90.

You will subtract 36 from 90 to find the measure of the missing angle.

w=90-36

w=54

The missing angle measures at 54^{\circ}.

4. \angle f and \angle e are complementary angles. \angle f measures at 66^{\circ}. What is the measure of \angle e?

e=123^{\circ}

e=90^{\circ}

e=44^{\circ}

e=24^{\circ}

The two angles are complementary, so they must have a sum of 90.

You will subtract 66 from 90 to find the measure of the missing angle.

e=90-66

e=24

The missing angle measures at 24^{\circ}.

5. Two angles ‘x and y' are complementary and one of them is 47^{\circ}. What is the size of the other angle?

x=33^{\circ}

x=43^{\circ}

x=133^{\circ}

x=313^{\circ}

The two angles are complementary, so they must have a sum of 90.

You will subtract 47 from 90 to find the measure of the missing angle.

x=90-47

x=43

The missing angle measures at 43^{\circ}.

6. Two angles ‘x and y' are complementary and one of them is 3^{\circ}. What is the size of the other angle?

x=87^{\circ}

x=3^{\circ}

x=93^{\circ}

x=177^{\circ}

The two angles are complementary, so they must have a sum of 90.

You will subtract 3 from 90 to find the measure of the missing angle.

x=90-3

x=87

The missing angle measures at 87^{\circ}.

## Complementary angles FAQs

What is the definition of complementary angles?

What’s the difference between complementary and supplementary angles?

Complementary angles add up to 90 degrees, or a right angle.
Supplementary angles add up to 180 degrees, or a straight line or straight angle.

Can three angles be complementary?

No, three angles will never be complementary, even if the sum of their measurement is 90^{\circ}. You will only see a pair of complementary angles.

Is there a formula for complementary angles?

\angle a+ \angle b=90^{\circ}; two angles are complementary if angle \, a and angle \, b equal 90^{\circ}.

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