1.1.4 Pairs of Angles

Key Objectives • Identify adjacent, vertical, complementary, and supplementary angles. • Find measures of pairs of angles. Key Terms • Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points. • A linear pair of angles is a pair of adjacent angles whose noncommon sides are opposite rays.

• Complementary angles are two angles whose measures have a sum of 90°. • Supplementary angles are two angles whose measures have a sum of 180°. • Vertical angles are two nonadjacent angles formed by two intersecting lines. Example 1 Identifying Angle Pairs

Two angles that have a common vertex and side, but do not overlap, are called adjacent angles. The figure to the left contains a pair of adjacent angles, ∠ BAC and ∠ CAD . Notice that ∠ BAD and ∠ BAC share a common vertex, point A , and a common side, ray AB , but those two angles are not adjacent angles because they overlap. A linear pair is a pair of adjacent angles that combine to form a straight angle. In a linear pair, the noncommon sides of the two adjacent angles are opposite rays. Five angles are named with numbers in this figure. For each given pair of angles, determine the type of pair by first considering whether they are adjacent. If they are adjacent, then consider whether they are a linear pair. ∠ 1 and ∠ 2 do not have a common vertex, so they are not adjacent. ∠ 4 and ∠ 5 have a common vertex and a common side. Furthermore, ∠ 4 and ∠ 5 do not overlap (i.e., they do not have any common interior points), so they must be adjacent. However, they are not a linear pair because their noncommon sides are not opposite rays. Therefore, ∠ 4 and ∠ 5 are adjacent only. ∠ 1 and ∠ 3 have a common vertex, a common side, and they do not overlap. So, they are adjacent. The noncommon sides of ∠ 1 and ∠ 3 are opposite rays. Therefore, ∠ 1 and ∠ 3 form a linear pair.

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