Key takeaway: Complementary angles are a pair of angles that together fill a square corner, with measures totaling 90 degrees.
Grade 77.G.B.5Geometry and Measurement
Definition of Complementary Angles
Complementary angles are a pair of angles that fit together to make a quarter turn. Their measures sum to \(90^{\circ}\). Each one is acute, because a pair adding to 90 leaves no room for an angle of 90 or more.
The two angles may sit side by side, sharing a vertex and a ray, as when a ray divides a corner of a picture frame. They may also be completely separate. Whether adjacent or not, the sum is what counts. One is called the complement of the other.
Formula and notation
For two angles named \(A\) and \(B\), the relationship reads
\[m\angle A + m\angle B = 90^{\circ}\]
so the partner of an angle measuring \(x^{\circ}\) is \((90 – x)^{\circ}\). A small square drawn in a corner marks a right angle.
Key facts
- Both members of a pair must be acute, each between 0 and 90 degrees.
- Complementary angles do not have to be next to each other.
- The two acute angles of any right triangle are always complementary angles.
- To find a missing partner, subtract the known angle from 90.
Complementary Angles example
Two angles measuring \((2x + 5)^{\circ}\) and \((3x – 10)^{\circ}\) are complementary. Find \(x\) and both angle measures.
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Set the sum equal to 90: \((2x + 5) + (3x – 10) = 90\).
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Combine: \(5x – 5 = 90\), so \(5x = 95\) and \(x = 19\).
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Angles: \(2(19) + 5 = 43\) and \(3(19) – 10 = 47\).
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Check: \(43 + 47 = 90\).
\(x = 19\), and the angles measure \(43^{\circ}\) and \(47^{\circ}\).
Common mistake
Watch out
Pedro is told an angle is \(64^{\circ}\) and says its complement is \(180 – 64 = 116^{\circ}\).
Why it is wrong
He used the total for a straight line. Complementary pairs sum to 90, so the answer is \(90 – 64 = 26\) degrees. A memory hook: the letter C comes first in the alphabet, just as 90 comes first when counting up to 180. Another check: a 64 degree angle looks wide, so its partner inside a square corner must be small, and 116 is far too big to fit.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Geometry and Measurement