Supplementary Angles Calculator
Enter one angle (in degrees or radians) to find its supplement, which is the angle that adds up to 180 degrees. You can also enter two angles to verify whether they form a supplementary pair. Switch between degrees and radians, and see the trigonometric relationships between supplementary angles.
Formula
Worked example
Given angle alpha = 60 degrees. Supplement = 180 - 60 = 120 degrees. Verification: 60 + 120 = 180. Trig check: sin(60) = sin(120) = 0.8660. cos(120) = -cos(60) = -0.5. tan(120) = -tan(60) = -1.7321.
What are supplementary angles?
Two angles are supplementary when their measures add up to exactly 180 degrees (or pi radians). The word "supplementary" comes from the Latin "supplementum," meaning something that completes or fills in. A classic example is a straight line: any two adjacent angles that together form a straight line are supplementary, because a straight line measures 180 degrees. If one angle measures 60 degrees, its supplement is 120 degrees, because 60 + 120 = 180. Either angle is called the supplement of the other.
How to find a supplementary angle
The formula is simple: supplement = 180 degrees - angle. To find the supplement of 75 degrees, calculate 180 - 75 = 105 degrees. In radians, the formula becomes supplement = pi - angle. To find the supplement of pi/4, calculate pi - pi/4 = 3pi/4. Because the result must be a positive angle, any angle greater than 180 degrees has no valid supplement. A right angle (90 degrees) is unique: its supplement is also 90 degrees, so two right angles always form a supplementary pair.
Supplementary angles in geometry and real life
Supplementary angles appear throughout geometry and everyday structures. Co-interior angles (also called same-side interior angles) formed when a transversal crosses two parallel lines always add to 180 degrees. Adjacent angles along a straight line form a linear pair, which is the most common supplementary pair in diagrams. In a cyclic quadrilateral (a four-sided figure inscribed in a circle), each pair of opposite angles is supplementary. Architects and engineers use supplementary angles when designing roof pitches, ramps, and braced structures, where knowing one angle immediately gives the other. The 60-and-120-degree pair that appears in equilateral-triangle geometry is a frequent example in structural design.
Trig identities for supplementary angles
If two angles alpha and beta are supplementary, so alpha + beta = 180 degrees, three trigonometric identities always hold. First, sin(alpha) = sin(beta): the sines are always equal. Second, cos(alpha) = -cos(beta): the cosines are equal in magnitude but opposite in sign. Third, tan(alpha) = -tan(beta): the tangents are equal in magnitude but opposite in sign. These identities are used to simplify expressions in trigonometry and calculus. For example, sin(150 degrees) = sin(30 degrees) = 0.5, and cos(150 degrees) = -cos(30 degrees) = -0.866.
Supplementary vs complementary angles
Supplementary angles sum to 180 degrees; complementary angles sum to 90 degrees. These are different concepts and are often confused. A 30-degree angle has a complement of 60 degrees (because 30 + 60 = 90) and a supplement of 150 degrees (because 30 + 150 = 180). A memory trick: "C" comes before "S" in the alphabet, and 90 comes before 180 numerically, so complementary goes with 90 and supplementary goes with 180.
Common supplementary angle pairs
| Angle | Supplement | Type | Notes |
|---|---|---|---|
| 1 deg | 179 deg | Acute / Obtuse | Near-straight |
| 30 deg | 150 deg | Acute / Obtuse | pi/6 and 5pi/6 |
| 45 deg | 135 deg | Acute / Obtuse | pi/4 and 3pi/4 |
| 60 deg | 120 deg | Acute / Obtuse | pi/3 and 2pi/3 |
| 90 deg | 90 deg | Right / Right | Linear pair of right angles |
| 120 deg | 60 deg | Obtuse / Acute | 2pi/3 and pi/3 |
| 135 deg | 45 deg | Obtuse / Acute | 3pi/4 and pi/4 |
| 150 deg | 30 deg | Obtuse / Acute | 5pi/6 and pi/6 |
| 179 deg | 1 deg | Obtuse / Acute | Near-straight |
Frequently tested supplementary angle pairs in geometry and trigonometry.
Frequently asked questions
What is the supplement of 90 degrees?
90 degrees. When you subtract 90 from 180 you get 90, so a right angle is its own supplement. Two right angles placed end to end form a straight line of 180 degrees.
Can two obtuse angles be supplementary?
No. An obtuse angle is greater than 90 degrees, so two obtuse angles would sum to more than 180 degrees. Supplementary pairs must consist of one acute and one obtuse angle, or two right angles (each exactly 90 degrees).
Can two acute angles be supplementary?
No. Two acute angles each measure less than 90 degrees, so their sum is less than 180 degrees. At least one angle in a supplementary pair must be at least 90 degrees.
What is the difference between supplementary and complementary angles?
Supplementary angles add up to 180 degrees; complementary angles add up to 90 degrees. A memory tip: "S" for supplementary and "S" for Straight (180-degree straight line), "C" for complementary and "C" for Corner (90-degree right-angle corner).
How do supplementary angles relate to parallel lines?
When a transversal crosses two parallel lines, it creates co-interior angles (also called same-side interior or consecutive interior angles) on the same side of the transversal. These co-interior angles are always supplementary, adding up to 180 degrees. This is a key theorem used to prove lines are parallel and to find missing angles in geometry problems.
Why is sin(angle) equal to sin(supplement)?
Because sine is positive in both the first and second quadrants of the unit circle, and supplementary angles fall on opposite sides of the y-axis at the same height. Algebraically, sin(180 - alpha) = sin(alpha), a direct consequence of the angle-subtraction identity for sine. Cosine reverses sign because cos(180 - alpha) = -cos(alpha), and tangent follows suit: tan(180 - alpha) = -tan(alpha).