460 CHAPTER 8 Geometry When two straight lines intersect, the nonadjacent angles formed are called vertical angles. In Fig. 8.14, 1 and 3 are vertical angles, and 2 and 4 are vertical angles. We can show that vertical angles have the same measure; that is, they are equal. For example, Fig. 8.14 shows that + = + = m m m m 1 2 180°. 2 3 180°. Since 2 has the same measure in both cases, m 1 must equal m 3. 2 1 3 4 Figure 8.14 1 2 3 4 5 6 7 8 Transversal l1 l2 Figure 8.15 Vertical Angles Vertical angles have the same measure. A line that intersects two different lines, l1 and l ,2 at two different points is called a transversal. Fig. 8.15 illustrates that when two parallel lines are cut by a transversal, eight angles are formed. Angles 3, 4, 5, and 6 are called interior angles, and angles 1, 2, 7, and 8 are called exterior angles. Eight pairs of supplementary angles are formed. Special names are given to the angles formed by a transversal crossing two parallel lines. We describe these angles below. Name Description Illustration Pairs of Angles Meeting Criteria Alternate interior angles Interior angles on opposite sides of the transversal 1 3 4 2 5 7 8 6 3 and 6 4 and 5 Alternate exterior angles Exterior angles on opposite sides of the transversal 1 3 4 2 5 7 8 6 1 and 8 2 and 7 Corresponding angles One interior and one exterior angle on the same side of the transversal 1 3 4 2 5 7 8 6 1 and 5 2 and 6 3 and 7 4 and 8 Parallel Lines Cut by a Transversal When two parallel lines are cut by a transversal, 1. alternate interior angles have the same measure. 2. alternate exterior angles have the same measure. 3. corresponding angles have the same measure. Example 7 Determining Angle Measures Fig. 8.16 shows two parallel lines cut by a transversal. If m 8 52°, = determine the measure of 1 through 7. m]8 5 528 l1 l2 1 2 4 5 7 6 3 Figure 8.16 When two parallel lines are cut by a transversal, we have the following angle measurement relationships.
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