Supplementary angles

Supplementary angles are two angles that add up to 180°. Each one is called the supplement of the other, so 130° and 50° are supplementary, and 50° is the supplement of 130°. To find the supplement of any angle, take it away from 180°.

What are supplementary angles?

Two angles are supplementary when their sizes add up to 180°, the angle on a straight line. That is the only condition. They can sit side by side, be far apart, or belong to different shapes:

• Adjacent supplementary angles share a vertex and a side, and together they make a straight line. A pair like this is called a linear pair.
• Non-adjacent supplementary angles just have the right total. A 70° angle in one corner of a page and a 110° angle in another are supplementary.

The definition is always about two angles. Three angles of 60° add up to 180°, but they are not called supplementary.

How to find the supplement of an angle

A few patterns fall out of the table. The supplement of an acute angle (under 90°) is obtuse (between 90° and 180°), and the other way round. A right angle is its own supplement. And only angles smaller than 180° have a supplement, because angles can't be negative.

Supplementary vs complementary angles

These two get mixed up more than anything else in angle facts. Complementary angles add up to 90°, a right angle. Supplementary angles add up to 180°, a straight line.

Two ways to remember which is which: C comes before S in the alphabet, just as 90 comes before 180. Or think C for corner and S for straight.

Where supplementary angles come up

Once you know to look for 180°, supplementary angles appear all over geometry. For every angle fact in one place, see angles.

• Angles on a straight line. Where two lines cross, angles next to each other are supplementary, while opposite ones are equal. See vertical angles.
• Parallel lines. When a line crosses two parallel lines, the two inside angles that sit on one side of it are supplementary. US textbooks call them same-side interior or consecutive interior angles. In the UK they are usually called co-interior angles.
• Parallelograms. Any two angles next to each other in a parallelogram are supplementary, because the opposite sides are parallel.
• Polygons. At each corner, the interior angle and the exterior angle sit on a straight line, so they are supplementary.
• Cyclic quadrilaterals. In a four-sided shape with all its corners on a circle, opposite angles are supplementary. This is one of the GCSE circle theorems.

Worked examples

In the US Common Core, grade 7 students use facts about supplementary, complementary, vertical and adjacent angles to write and solve equations for unknown angles. These examples build up to that.

Example 1: one angle is four times the other

Two angles are supplementary, and one is four times the size of the other. Call the smaller angle x, so the larger is 4x:

The angles are 36° and 144°. Check: 36 + 144 = 180.

Example 2: expressions in x

Angles of (2x + 15)° and (3x − 5)° form a linear pair. Find x and both angles.

So x = 34, and the angles are 2(34) + 15 = 83° and 3(34) − 5 = 97°. They add up to 180°, as they should.

Example 3: 20° more than its supplement

An angle is 20° bigger than its supplement. Call the angle x. Its supplement is 180 − x, so:

The angle is 100° and its supplement is 80°, which is 20° smaller.

Example 4: supplement and complement together

The supplement of an angle is three times its complement. Find the angle.

Call the angle x. Its supplement is 180 − x and its complement is 90 − x:

The angle is 45°. Check: its supplement is 135°, its complement is 45°, and 135 = 3 × 45. Equations like these are covered in the Algebra course.

Example 5: a parallelogram

One angle of a parallelogram is 68°. The angles next to it are its supplements, 180 − 68 = 112°, and the angle opposite is equal to it. So the four angles are 68°, 112°, 68° and 112°, which add up to 360°.

Common mistakes

• Using 90° instead of 180°. That is complementary. Supplementary means a straight line, 180°.
• Thinking supplementary angles must touch. Any two angles adding to 180° are supplementary, wherever they are.
• Calling three angles supplementary. The word is for pairs only.
• Assuming two angles on a diagram make a straight line. Only use 180° if the line is marked or stated to be straight.
• Finding x and stopping. If the question asks for the angles, substitute x back in.

To practise angle facts mixed with other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 or Geometry, Algebra 2 & SAT Math paper. In the UK, the same practice opens as GCSE Maths Foundation.

Read next

Maths · GeometryComplementary angles: definition, examples and problemsWhat complementary angles are, how to find the complement of an angle, how they differ from supplementary angles, and how to solve problems with algebra.