Complementary angles are two angles that add up to 90°. Supplementary angles are two angles that add up to 180°. These are foundational geometry terms, and once you know the two numbers, most problems involving them become straightforward.
Complementary Angles: Adding to 90°
Two angles are complementary when their measures sum to exactly 90 degrees. They don’t need to be next to each other or part of the same shape. A 30° angle and a 60° angle are complementary whether they’re on opposite sides of a worksheet or sitting side by side forming a right angle. The only requirement is that their values add up to 90.
The most common place you’ll see complementary angles is inside a right angle. If a line splits a right angle into two smaller angles, those two pieces are always complementary. Think of the corner of a book tilted so that one angle is 25°. The remaining sliver is 65°, and together they complete the 90° corner.
To find an angle’s complement, subtract it from 90. If you have a 40° angle, its complement is 50°. If you have a 72° angle, its complement is 18°. Any angle greater than 90° has no complement, because you’d need a negative number to reach 90.
Supplementary Angles: Adding to 180°
Two angles are supplementary when their measures sum to exactly 180 degrees. The classic example is a straight line. When a ray extends from a point on a straight line, it creates two angles on either side. Those two angles are always supplementary because the straight line itself represents 180°.
To find an angle’s supplement, subtract it from 180. A 110° angle has a supplement of 70°. A 45° angle has a supplement of 135°. Unlike complements, supplementary angles can include obtuse angles (anything between 90° and 180°), which makes them useful in a wider range of geometry problems.
How to Tell Them Apart
A simple memory trick: the “c” in complementary comes before “s” in supplementary in the alphabet, and 90 comes before 180. Complementary goes with the smaller number (90°), supplementary with the larger (180°).
Another approach: think of “corner” for complementary (a right-angle corner is 90°) and “straight” for supplementary (a straight line is 180°).
Solving Problems With These Angles
Most textbook problems give you one angle and ask you to find its complement or supplement. The math is always the same: subtract from 90 for a complement, subtract from 180 for a supplement.
Here’s where it gets slightly more interesting. Suppose you’re told two complementary angles are in a 2:1 ratio. You know they add to 90°, so the larger angle is 60° and the smaller is 30°. The same logic applies to supplementary angles. If two supplementary angles are in a 3:1 ratio, divide 180 into four equal parts (45° each), giving you 135° and 45°.
You’ll also encounter problems where two angles are described with variables. If one angle is x and its supplement is 3x, you set up the equation x + 3x = 180, solve for x = 45, and the two angles are 45° and 135°. The same structure works for complementary angles, just swap 180 for 90.
Can Two Angles Be Both?
Only in one specific case. If two angles are each exactly 45°, they are complementary (45 + 45 = 90). If two angles are each exactly 90°, they are supplementary (90 + 90 = 180). But no pair of angles can be both complementary and supplementary at the same time, because that would require the same two numbers to add up to both 90 and 180.
It’s also worth noting that complementary and supplementary relationships always involve exactly two angles. Three angles that add to 180° aren’t called supplementary in standard geometry (though the interior angles of a triangle do always sum to 180°).
Where These Show Up in Geometry
Complementary angles appear whenever right angles are involved. The two acute angles inside every right triangle are complementary, since the three interior angles of any triangle sum to 180° and the right angle already accounts for 90°. This relationship is the basis for many trigonometry identities you’ll encounter later.
Supplementary angles come up constantly with parallel lines. When a line crosses two parallel lines, it creates pairs of angles on the same side that are supplementary. They also appear in quadrilaterals: consecutive angles in a parallelogram are always supplementary. Any time you see a straight line or a 180° relationship in a diagram, supplementary angles are at work.