Supplementary Angles
In short: Supplementary angles are two angles whose measures add to exactly 180 degrees, the size of a straight line, and they may be separate or sit side by side.
7th GradeGeometry7.G.B.5
What is Supplementary Angles?
When two angles together measure \(180^{\circ}\), they are supplementary angles. Fit them next to each other along one line and they unfold into a perfectly straight edge, like a door swung wide open.
If one angle has measure \(x\), the other has measure \(180^{\circ} – x\). A pair can mix an acute angle with an obtuse one, or consist of two right angles, but two acute angles can never be supplementary because their total falls short of a half turn.
How to spot it
- A diagram shows a ray leaving a straight line, creating two angles on the same side of that line.
- The wording says the angles add to 180 degrees, or that one is the supplement of the other.
- Two algebraic expressions describe angles that together make a straight angle.
- One obtuse angle is known and you must find the acute angle that completes the line.
The method
- State that the two measures sum to \(180^{\circ}\) and write the equation.
- If one angle is known, subtract it from \(180^{\circ}\).
- If both are expressions, collect like terms, then solve for the variable.
- Substitute back, find each angle and confirm that their total is \(180^{\circ}\).
Supplementary Angles: solved example
Problem
Two supplementary angles measure \((3x + 12)^{\circ}\) and \((5x – 8)^{\circ}\). Find \(x\) and each angle.
Solution
Their sum is a straight angle: \(3x + 12 + 5x – 8 = 180\), so \(8x + 4 = 180\) and \(x = 22\).
The angles are \(3(22) + 12 = 78\) and \(5(22) – 8 = 102\); indeed \(78 + 102 = 180\). They measure \(78^{\circ}\) and \(102^{\circ}\).
Try it
Question 1
Find the supplement of a \(127^{\circ}\) angle.
Show the answer
\(180 – 127 = 53\), so \(53^{\circ}\).
Question 2
Two supplementary angles are in the ratio 2 to 7. How large is each?
Show the answer
\(2k + 7k = 180\) gives \(k = 20\), so the angles are \(40^{\circ}\) and \(140^{\circ}\).
Supplementary Angles vs. Complementary Angles
Supplementary pairs total \(180^{\circ}\) and straighten into a line; complementary pairs total \(90^{\circ}\) and square into a corner. Another close cousin is the linear pair: adjacent supplementary angles that touch along a line. Every linear pair is supplementary, but supplementary angles need not touch.
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Whole dictionary: A–Z · 7th Grade · Geometry