Geometry Problem Solver What are supplementary angles?
Two angles are supplementary when their measures add up to exactly 180°. They do not have to touch. To find the supplement of any angle, subtract it from 180°: the supplement of 55° is 125°.
They do not have to be next to each other
This is the detail that costs marks. A 40° angle drawn on one page and a 140° angle drawn on another are still supplementary, because supplementary describes a relationship between two measures, not an arrangement on the page.
When they are adjacent and share a side, the pair forms a straight line and is called a linear pair. Every linear pair is supplementary. Not every supplementary pair is a linear pair.
Finding a missing angle
Subtract from 180°, and that is the whole method. If two angles are supplementary and one is 112°, the other is 180° − 112° = 68°.
When the problem gives you an expression instead of a number, set the sum equal to 180° and solve. If the angles are x and 3x, then x + 3x = 180°, so 4x = 180° and x = 45°, making the two angles 45° and 135°.
Where they turn up
On a straight line, any ray drawn from a point creates a supplementary pair. In a parallelogram, adjacent angles are supplementary. When a transversal cuts a pair of parallel lines, the co-interior angles, sometimes called same-side interior angles, are supplementary too, and that fact is what most parallel-line problems are really testing.
Supplementary against complementary
| Supplementary | Complementary | |
|---|---|---|
| Add up to | 180° | 90° |
| Forms | A straight line, when adjacent | A right angle, when adjacent |
| Find the partner | 180° − angle | 90° − angle |
| Can both be acute? | No, at most one | Yes, both always are |
Questions
Can two obtuse angles be supplementary?
No. Both would be over 90°, so their sum would exceed 180°. In any supplementary pair either one angle is acute and the other obtuse, or both are exactly 90°.
What is the supplement of 90°?
90°. It is the only angle that is its own supplement, and the pair happens to be complementary in nothing but name, since 90° + 90° = 180°.
How do I remember which is which?
C comes before S in the alphabet and 90 comes before 180. Complementary is the smaller pair, supplementary the larger.
More geometry reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
