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K12 LibreTexts

1.14: Complementary Angles

  • Page ID
    2128
  • 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.pngFigure \(\PageIndex{1}\)

    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.pngFigure \(\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:

    1. \(4^{\circ}\)
    2. \(89^{\circ}\)
    3. \(54^{\circ}\)
    4. \(32^{\circ}\)
    5. \(27^{\circ}\)
    6. \((x+y)^{\circ}\)
    7. \(z^{\circ}\)

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

    f-d_ddbfde705abde0cdc132cdfcd782bec102f8b34c4b23a3ab4f4862ca+IMAGE_TINY+IMAGE_TINY.pngFigure \(\PageIndex{3}\)
    1. Name two complementary angles.
    1. If \(m \angle INJ=63^{\circ}\), find \(m \angle KNJ\).

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

    1. Complementary angles add up to \(180^{\circ}\).
    2. 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

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