Angle pairs show up in nearly every geometry problem, and three pairs matter most: complementary, supplementary, and vertical angles. Each pair follows one simple rule. You can check your work with the free tools on Calc Solver.
Complementary Angles Add to 90 Degrees
Two angles are complementary when they add up to 90 degrees, the angle of a right angle. If one angle is 30 degrees, its complement is 60 degrees, because 30 plus 60 is 90. To find a complement, subtract the known angle from 90.
Supplementary Angles Add to 180 Degrees
Two angles are supplementary when they add up to 180 degrees, the angle of a straight line. If one angle is 110 degrees, its supplement is 70 degrees, because 110 plus 70 is 180. To find a supplement, subtract the known angle from 180.
Vertical Angles Are Always Equal
When two straight lines cross, the angles opposite each other are called vertical angles, and they are always equal. If one angle at the crossing is 70 degrees, the angle opposite it is also 70 degrees, while the two angles beside it are each 110 degrees, since neighbors on a straight line are supplementary.
How to Spot Each Pair in a Diagram
Look for a right-angle mark to find complementary pairs, a straight line to find supplementary pairs, and an X-shaped crossing to find vertical pairs. Naming the pair first makes the computation obvious, because each pair gives you exactly one equation to solve.
Worked Example
Two lines cross, and one angle measures 125 degrees. The vertical angle opposite it is also 125 degrees. Each neighboring angle is supplementary to it, so each is 180 minus 125, which is 55 degrees. All four angles are now known from a single measurement.
Common Mistakes to Avoid
First, mixing up complementary (90) and supplementary (180). A memory aid is that “complementary” and “corner” both start with C, and a corner is 90 degrees. Second, assuming angles are a pair just because they look close: the rule depends on the diagram, not on appearance alone.
Practice Problems to Try
Find the complement of 42 degrees, the supplement of 135 degrees, and all four angles when two lines cross with one angle of 80 degrees.