1.14: Complementary Angles

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Two angles that add up to 90 degrees.

Two angles are complementary if they add up to \(90^{\circ}\). Complementary angles do not have to be congruent or adjacent.

What if you were given two angles of unknown size and were told they are complementary? How would you determine their angle measures?

Example \(\PageIndex{1}\)

Find the measure of an angle that is complementary to \(\angle ABC\) if \(m \angle ABC\) is \(82^{\circ}\).

Solution

\(90^{\circ}−82^{\circ}=8^{\circ}\).

Example \(\PageIndex{2}\)

Find the measure of an angle that is complementary to \(\angle ABC\) if \(m\angle ABC\) is \(12^{\circ}\).

Solution

\(90^{\circ}−12^{\circ}=78^{\circ}\).

Example \(\PageIndex{3}\)

The two angles below are complementary. \(m \angle GHI=x\). What is \(x\)?

f-d_1eda082f61bef4c6adb0a930c06c64f8f79eb51b92bfa5c7f4598fa4+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{1}\)

Solution

Because the two angles are complementary, they add up to \(90^{\circ}\). Make an equation.

\(x+34^{\circ}=90^{\circ}\)

\(x=56^{\circ}\)

Example \(\PageIndex{4}\)

The two angles below are complementary. Find the measure of each angle.

f-d_19f0fa04b5e1b3458adfcd1a8c9d627795351966e4fe3291a377a89e+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{2}\

Solution

The two angles add up to \(90^{\circ}\). Make an equation.

\((8r+9)+(7r+6)=90^{\circ}\)

\((15r+15)=90^{\circ}\)

\(15r=75^{\circ}\)

\(r=5^{\circ}\)

However, you need to find each angle. Plug r back into each expression.

(m\angle GHI=8(5^{\circ})+9^{\circ}=49^{\circ}\)

\(m \angle JKL=7(5^{\circ})+6^{\circ}=41^{\circ}\)

Example \(\PageIndex{5}\)

Find the measure of an angle that is complementary to \(\angle MRS\) if \(m \angle MRS\) is \(70^{\circ}\).

Solution

Because complementary angles have to add up to \(90^{\circ}\), the other angle must be \(90^{\circ}−70^{\circ}=20^{\circ}\).

Review

Find the measure of an angle that is complementary to \(\angle ABC\) if \(m \angle ABC\) is:

\(4^{\circ}\)
\(89^{\circ}\)
\(54^{\circ}\)
\(32^{\circ}\)
\(27^{\circ}\)
\((x+y)^{\circ}\)
\(z^{\circ}\)

Use the diagram below for exercises 8-9. Note that \(\overline{NK} \perp \overleftrightarrow{IL}\).

f-d_ddbfde705abde0cdc132cdfcd782bec102f8b34c4b23a3ab4f4862ca+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{3}\)

Name two complementary angles.

If \(m \angle INJ=63^{\circ}\), find \(m \angle KNJ\).

For 10-11, determine if the statement is true or false.

Complementary angles add up to \(180^{\circ}\).
Complementary angles are always \(45^{\circ}\).

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.7.

Resource

Vocabulary

Term	Definition
complementary angles	Two angles are *complementary* if they add up to \(90^{\circ}\).

Additional Resource

Interactive Element

Video: Complementary, Supplementary, and Vertical Angles

Activities: Complementary Angles Discussion Questions

Study Aids: Angles Study Guide

Practice: Complementary Angles

Real World: Complementary Angles