Complementary angles are any two angles whose sum is 90°. In Figure 3, because ∠ ABC is a right angle, m ∠1 + m ∠2 = 90°, so ∠1 and ∠2 are complementary.
Figure 3 Adjacent complementary angles.
Complementary angles do not need to be adjacent. In Figure 4, because m ∠3 + m ∠4 = 90°, ∠3, and ∠4, are complementary.
Figure 4 Nonadjacent complementary angles
Example 1: If ∠5 and ∠6 are complementary, and m ∠5 = 15°, find m ∠6.
Because ∠5 and ∠6 are complementary,
Theorem 8: If two angles are complementary to the same angle, or to equal angles, then they are equal to each other.
Refer to Figures 5 and 6. In Figure 5, ∠ A and ∠ B are complementary. Also, ∠ C and ∠ B are complementary. Theorem 8 tells you that m ∠ A = m ∠ C. In Figure 6, ∠ A and ∠ B are complementary. Also, ∠ C and ∠ D are complementary, and m ∠ B = m ∠ D. Theorem 8 now tells you that m ∠ A = m ∠ C.
Figure 5 Two angles complementary to the same angle
Figure 6 Two angles complementary to equal angles
Supplementary angles
Supplementary angles are two angles whose sum is 180°. In Figure , ∠ ABC is a straight angle. Therefore m ∠6 + m ∠7 = 180°, so ∠6 and ∠7 are supplementary.
Figure 7 Adjacent supplementary angles.
Theorem 9: If two adjacent angles have their noncommon sides lying on a line, then they are supplementary angles.
Supplementary angles do not need to be adjacent (Figure 8).
Figure 8 Nonadjacent supplementary angles.
Because m ∠8 + m ∠9 = 180°, ∠8 and ∠9 are supplementary.
Theorem 10: If two angles are supplementary to the same angle, or to equal angles, then they are equal to each other.