What Are Supplementary Angles? Supplementary Angles Definition
The adjective supplementary describes a special relationship between two angles. According to the supplementary angles definition, two angles are supplementary when their measures add up to exactly 180° (or π if you are using radians). In other words, if they were adjacent, they would form a straight line.
α + β = 180° = π rad
β = 180° − α
Keep in mind the essential properties of supplementary angles:
- Only two angles that sum up to 180° (π) are supplementary. Three or more angles can, of course, add up to 180° (that happens in the triangle angle calculator), but they are not called supplementary.
- Your two supplementary angles cannot be both obtuse or both acute — there are only two options:
- One angle is acute, and the other is obtuse, or
- Both of them are right angles (90° + 90° = 180°).
How to Find Supplementary Angles?
Let's consider two probable scenarios — you are searching for a supplementary angle to your given angle, or you have two angles and you wonder if they are supplementary. You can solve both questions with this supplementary angles calculator.
1st scenario: What is the supplementary angle?
Subtract your given angle from 180°:
supplementary angle = 180° − angle
or in radians: supplementary angle = π − angle
2nd scenario: Are these two angles supplementary?
Check if their sum equals 180°:
- angle₁ + angle₂ = 180° (π) — the angles are supplementary
- angle₁ + angle₂ ≠ 180° (π) — the angles are not supplementary
So, for example, are these two angles supplementary?
- 30° and 150° are supplementary (as they add up to 180°) ✓
- 2π/3 and π/3 are supplementary (as they sum up to π) ✓
- but 60° and 140° are not supplementary ✗
Check your calculations with our supplementary angles calculator!
Adjacent Supplementary Angles
Searching for adjacent supplementary angles in geometry is so frequent and natural that you may not even realize you are doing so! You are using the definition of the supplementary angle every time you have two lines or line segments that intersect (a linear pair), as they form pairs of adjacent angles that are supplementary.
- Adjacent supplementary angles share a common vertex and a common side, and their non-common sides form a straight line (180°). This is known as a linear pair.
- Non-adjacent supplementary angles are two separate angles that simply sum to 180° without sharing a side.
Supplementary Angles Relationships
Supplementary angles appear throughout geometry. Some of the most common relationships include:
- Linear pair: When two lines intersect, each pair of adjacent angles is supplementary.
- Co-interior (consecutive interior) angles: When a transversal crosses two parallel lines, the co-interior angles on the same side are supplementary.
- Cyclic quadrilateral: Opposite angles of a quadrilateral inscribed in a circle are supplementary.
- Straight angle: Any angles that together form a straight line add up to 180°.
Don't forget to check out the twin brother of this calculator — the complementary angles calculator, where the two angles sum up to 90° instead of 180°.
Complementary vs. Supplementary Angles
| Property | Complementary | Supplementary |
|---|---|---|
| Sum equals | 90° (π/2 rad) | 180° (π rad) |
| Example | 30° and 60° | 30° and 150° |
| On a straight line | No | Yes — a linear pair |
| Can both be right angles? | No | Yes — 90° + 90° |
Measurement Systems
This calculator supports two measurement systems:
- Degrees (°) — US/Imperial system: The most common system in everyday use. A full rotation is 360°, and a straight angle is 180°. Supplementary angles sum to 180°.
- Radians (rad) — Metric/SI system: The standard unit in mathematics and physics. A full rotation is 2π rad, and a straight angle is π ≈ 3.1416 rad. Supplementary angles sum to π rad.
Conversion: 1 radian = 180°/π ≈ 57.2958° | 1° = π/180 ≈ 0.017453 rad
Common Supplementary Angle Pairs
| Angle α (degrees) | Angle α (radians) | Supplement (degrees) | Supplement (radians) |
|---|---|---|---|
| 0° | 0 | 180° | π ≈ 3.1416 |
| 30° | π/6 ≈ 0.5236 | 150° | 5π/6 ≈ 2.6180 |
| 45° | π/4 ≈ 0.7854 | 135° | 3π/4 ≈ 2.3562 |
| 60° | π/3 ≈ 1.0472 | 120° | 2π/3 ≈ 2.0944 |
| 90° | π/2 ≈ 1.5708 | 90° | π/2 ≈ 1.5708 |
| 120° | 2π/3 ≈ 2.0944 | 60° | π/3 ≈ 1.0472 |
| 135° | 3π/4 ≈ 2.3562 | 45° | π/4 ≈ 0.7854 |
| 150° | 5π/6 ≈ 2.6180 | 30° | π/6 ≈ 0.5236 |
| 180° | π ≈ 3.1416 | 0° | 0 |
Frequently Asked Questions
- Can an angle be supplementary to itself?
- Yes — 90° is the only angle that is self-supplementary (90° + 90° = 180°).
- Can two acute angles be supplementary?
- No. Two acute angles (each less than 90°) sum to less than 180°, so they cannot be supplementary.
- Can two obtuse angles be supplementary?
- No. Two obtuse angles (each greater than 90°) sum to more than 180°, so they cannot be supplementary either.
- What is the supplement of 0°?
- The supplement of 0° is 180°. Conversely, the supplement of 180° is 0°.
- Are supplementary angles always adjacent?
- No. Supplementary angles only need to sum to 180°. When they are adjacent, they form a linear pair on a straight line, but they can also be non-adjacent.