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Complementary Angles

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Complementary Angles

What if you knew that two angles together made a right angle? After completing this Concept, you'll be able to use what you know about complementary angles to solve problems about these angles.

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CK-12 Foundation: Chapter1ComplementaryAnglesA

James Sousa: Complementary Angles


Two angles are complementary when they add up to 90^\circ . Complementary angles do not have to be congruent to each other, nor do they have to be next to each other.

Example A

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

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

x + 34^\circ = 90^\circ\\x = 56^\circ

Example B

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

Again, the two angles add up to 90^\circ . Make an equation.

8r + 9^\circ + 7r+ 5^\circ & = 90^\circ\\15r + 14^\circ & = 90^\circ\\15r & = 76^\circ\\r & \approx 5.1^\circ

However, this is not what the question asks for. You need to plug r back into each expression to find each angle.

For m \angle GHI : 8(5.1^\circ) + 9^\circ = 49.8^\circ , so m \angle GHI \approx 49.8^\circ .

For m \angle JKL : 7(5.1^\circ) + 5^\circ = 40.7^\circ , so m \angle JKL \approx 40.7^\circ .

Example C

Name one pair of complementary angles in the diagram below.

One example is   \angle INJ and  \angle JNK .

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter1ComplementaryAnglesB

Guided Practice

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

  1. 45^\circ
  2. 82^\circ
  3. 19^\circ
  4. z^\circ


1. 45^\circ

2. 8^\circ

3. 71^\circ

4. 90-z^\circ

Interactive Practice

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

  1. 3^\circ
  2. 82^\circ
  3. 51^\circ
  4. 30^\circ
  5. 22^\circ
  6. (x+y)^\circ
  7. x^\circ

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

  1. If m\angle INJ = 60^\circ , find m\angle KNJ .
  2. If m\angle INJ = 70^\circ , find m\angle KNJ .

For 10-15, determine if the statement is true or false.

  1. Complementary angles add up to 180^\circ .
  2. Complementary angles are always 45^\circ .
  3. Complementary angles are always next to each other.
  4. Complementary angles add up to 90^\circ .
  5. Two angles that make a right angle are complementary.
  6. The two non-right angles in a right triangle are complementary.

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