Two angles are complementary when their measures add to 90°. An angle of 30° and an angle of 60° are complementary, whether they appear next to each other in a drawing or in separate figures. The definition concerns their measures, not their position.
How do you find the complement of an angle?
Subtract the known angle from 90°. If the measures are α and β, their relationship is:
For a known angle of 20°, the missing angle is 70°:
Finding the third angle of a right triangle
A triangle has a right angle and an angle of 20°. Find its remaining interior angle β. The triangle's three angles total 180°.
Write the sum using the two known angles and the unknown angle.
Remove the right angle from the total. The two remaining angles must add to 90°, so subtract 20° to isolate β.
Check the answer in the original three-angle sum, rather than only in the shortened equation.
The missing angle is 70°. The 20° and 70° angles are complementary, while all three angles together form the triangle's 180° total.
Check the answer by adding the two measures: 20° + 70° = 90°. Subtracting 90° from 20° would reverse the calculation and produce a negative value, which is not the ordinary interior angle being sought here.
Must complementary angles be adjacent?
No. Adjacent angles share a vertex and one side, and their interiors do not overlap. If an interior ray divides a right angle, the two resulting angles are both adjacent and complementary. But adjacency alone does not make a pair complementary. Two adjacent angles measuring 40° and 80° add to 120°.
Conversely, two separate angles measuring 25° and 65° remain complementary. This distinction is useful when a diagram spreads the angles across different parts of a construction. Check the measures rather than assuming the relationship from their position.
Complementary, supplementary or a full turn?
Complementary angles add to 90°; supplementary angles add to 180°. A pair adding to 360° completes a full turn and is sometimes described as explementary. For a 20° angle, the complement is 70°, the supplement is 160°, and the angle completing a full turn is 340°. These are three different questions.
Measuring a pair with a protractor
Place the centre of the protractor at the vertex and align its zero line with one side. Read the scale that starts at zero on that side. A double-scale protractor may show both 20° and 160° at the same mark, so use the opening of the angle to decide which reading fits.
Measure both angles before adding them. A drawing may be illustrative rather than accurately scaled, and a printed line has thickness. Exact numerical measures or a marked right angle are stronger evidence than an approximate measurement.
Can every angle have a positive complement?
For two positive angles in this setting, each must be less than 90°. A 90° angle leaves 0°, a boundary case only if zero angles are admitted. An angle greater than 90° has no positive complement. In a right triangle, the two remaining interior angles are complementary because the triangle's three angles total 180°. See also angle types and measurement.
