Complementary and Supplementary Angles

Sums of 90° and 180°

If two angles are complementary, then the sum of their measures is 90°. In the given diagram, angles CBD and ABD together form right angle ABC. Recall from geometry that perpendicular lines (like the x- and y-axes) form right angles, which measure 90°. Therefore, the sum of the measures of the angles is 90°:

In the equation above, and represent the measures of and , respectively.

The problem states that and are complementary, so the sum of their measures is 90°.

+=90°

Substitute =31° into the equation and solve for .

Complementary angles have measures that sum to 90°. This problem expresses the degree measure in radians, so you should begin by converting 90° to radians as well.

Problem 1.24 explains this step—how to convert from degrees to radians.

Therefore, complementary angles measured in radians have a sum of π/2.

Substitute = π/11 into the equation

Multiply both sides of the equation by 22, the least common multiple of 2 and 11, to eliminate the fractions.

Reduce the fractions to lowest terms and solve for m∠1.

The complement of π/11 is 9π/22.

The small square located at Y indicates that , so is a right angle. According to a geometric theorem, the acute angles of a right triangle are complementary. Recall that acute angles measure less than 90° (or π/2 radians), so in this triangle, and are acute.

The measures of the angles in a triangle add up to 180°. If you subtract the 90° right angle from this total, the two remaining acute angles have to add up to the remaining 90°, so they are complementary. For more details, see Problem 7.28 in The Humongous Book of Geometry Problems.

According to the diagram, = π/6. Substitute this value into the equation above and solve for .

Express the fractions in terms of the least common denominator, 6.

Reduce the fraction to lowest terms.

You conclude that = π/3.

A pair of angles is supplementary if their measures have a sum of 180°. In the given diagram, angles 3 and 4 combine to create a straight angle—the line . Lines are also described as “straight angles” and have a measure of 180°. Therefore, angles 3 and 4 are supplementary, and their measures have a sum of 180°.

According to Problem 1.35, angles 3 and 4 are supplementary. Therefore, their measures have a sum of 180°.

Substitute = 75° into the equation and solve for .

The measures of supplementary angles have a sum of π.

x

180° = πradians

Substitute = 5π/7 into the equation and solve for .

Arrowheads in the middle of lines indicate that the lines are parallel to each other.

According to a geometric theorem, when two parallel lines are intersected by a third, non-parallel line (called the transversal), same-side interior angles are supplementary. Same-side interior angles are defined as two angles on the same side of the transversal located between the parallel lines.

In other words, angles 1 and 2 are same-side interior angles because they are both on the same side of p (both are above the line) and both are located between parallel lines l and m. Therefore, the angles are supplementary. Note that the problem presents an angle measured in radians, so rather than set the sum of the supplementary angles equal to 180°, you should set it equal to π.

Substitute = π/5 into the equation and solve for . Note that you will need to use common denominators to simplify the result.

Because and combine to form a straight line—which measures π radians—you can conclude that the angles are supplementary. Thus, the sum of their measures is equal to π.

Solve for x.

The answer is also correct.