Key takeaway: Supplementary angles are a pair of angles that together fill a straight line, with measures totaling 180 degrees.
Grade 77.G.B.5Geometry and Measurement
Definition of Supplementary Angles
Supplementary angles are a pair of angles whose measures sum to \(180^{\circ}\), half a full turn. If one is obtuse, the other must be acute; if one is a right angle, so is the other. Think of a door swinging open: the angle on one side of the door and the angle on its other side add up to a flat wall.
When the two angles sit side by side along a straight line, they form a linear pair. But supplementary angles do not have to touch. An angle of \(112^{\circ}\) in one corner of a roof and an angle of \(68^{\circ}\) in a faraway window frame are supplementary, since \(112 + 68 = 180\).
Formula and notation
For two angles named \(A\) and \(B\), the relationship reads
\[m\angle A + m\angle B = 180^{\circ}\]
so the supplement of an angle measuring \(x^{\circ}\) is \((180 – x)^{\circ}\). Solving for a missing angle is a one-step subtraction from 180.
Key facts
- Any two angles that form a linear pair are supplementary.
- Two right angles are a pair that sums to 180, since \(90 + 90 = 180\).
- Supplementary angles may be far apart; they need no shared vertex or side.
- Co-interior angles between parallel lines are supplementary too.
Supplementary Angles example
Two supplementary angles measure \((5x + 12)^{\circ}\) and \((4x + 6)^{\circ}\). Find the value of \(x\) and the size of each angle.
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Their sum is 180: \((5x + 12) + (4x + 6) = 180\).
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Combine like terms: \(9x + 18 = 180\), so \(9x = 162\) and \(x = 18\).
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Angles: \(5(18) + 12 = 102\) and \(4(18) + 6 = 78\).
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Check: \(102 + 78 = 180\).
\(x = 18\); the angles are \(102^{\circ}\) and \(78^{\circ}\).
Common mistake
Watch out
Zoe sees two angles of \(112^{\circ}\) and \(68^{\circ}\) in different parts of a diagram and says they cannot be supplementary because they do not touch.
Why it is wrong
Touching is not part of the definition. Only the total matters: \(112 + 68 = 180\), so they are supplementary wherever they sit. When the angles are adjacent along a line, they are a linear pair, which is a special case.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Geometry and Measurement