Complementary Angles

Two angles that make a right angle

A right angle is a quarter turn, 90°. It is the corner of a square, and it is marked with a small square drawn in the corner instead of a curve.

Two angles are complementary when their sizes add up to 90°. Each angle is then called the complement of the other. The pair most often sits side by side, sharing one arm, so that together they fill a right angle exactly.

A right angle, 90°, with the small square in the corner.

Split a right angle with a ray

Draw a ray from the vertex into the corner, between the two arms. It cuts the right angle into two angles side by side. Suppose the lower one measures 35°.

The two angles have no gap between them and do not overlap, so together they make the whole right angle. The upper angle is what is left of the 90°: 90 − 35 = 55°. Check: 35 + 55 = 90. So 35° and 55° are complementary, and 55° is the complement of 35°.

Wherever the ray points inside the corner, the same is true. Swing it lower and the lower angle shrinks while the upper one grows by exactly the same amount, so the two always add up to 90°.

The base arm lies along 0. Ray A crosses the scale at 35 and ray B at 90, so B makes a right angle with the base arm. The angle from the base arm to A is 35°, and the angle from A to B is 90 − 35 = 55°.

Finding the missing angle

Suppose one angle of a complementary pair is 35° and the other is unknown. Call the unknown angle a. The pair adds up to 90, so a + 35 = 90. Take 35 from both sides: a = 90 − 35, so a = 55°.

In short, subtract the angle you know from 90. The complement of 72° is 90 − 72 = 18°, and the complement of 18° is 72°. Check each answer by adding the pair back up to 90.

Only an angle smaller than 90° has a complement. An angle of 90° leaves nothing to fill, and an angle bigger than 90° is already more than a right angle.

Two unknown angles

Two complementary angles differ by 40°. Call the smaller one x. The larger one is 40° more, x + 40. Together they make 90, so x + x + 40 = 90. That gives 2x + 40 = 90, then 2x = 50, and x = 25.

So the smaller angle is 25° and the larger is 25 + 40 = 65°. Check both facts: 25 + 65 = 90, and 65 − 25 = 40.

A right angle, not a straight line

Angles on a straight line add up to 180°, a half turn. Two angles that add up to 180° are called supplementary. A complementary pair fills a right angle, which is half of a straight line: 90°.

The usual mistake is to subtract from 180 instead of 90. The complement of 35° is 90 − 35 = 55°. 180 − 35 = 145° is the angle that finishes a straight line, not a right angle.

The other mistake is to assume the two angles are equal. They are equal only when each is 45°, because 45 + 45 = 90. The complement of 35° is 55°, not 35°.

A straight angle is 180°, two right angles. A complementary pair fills only one of them.

Perpendicular lines

Two straight lines that cross at a right angle are perpendicular, written with the sign ⊥: PQ ⊥ RS says that PQ and RS are perpendicular.

Where two perpendicular lines cross, all four angles are right angles. One of them is 90°. The angle beside it on the same straight line is 180 − 90 = 90°, and the next angle round is 90° for the same reason, and so on all the way round the crossing. A ray drawn between two of the arms splits one of those right angles into a complementary pair.