3.2 Transversals and Angles CHAPTER 3.2 Transversals and Angles Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions A transversal is a line that intersects two or more coplanar lines at different points. The figure below shows the eight angles formed by a transversal t and two lines / and m. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions Angles 3, 4, 5, and 6 lie between l and m. They are interior angles. Angles 1, 2, 7, and 8 lie outside of l and m. They are exterior angles. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions Alternate interior angles are nonadjacent interior angles that lie on opposite sides of the transversal. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions Same-side interior angles are interior angles that lie on the same side of the transversal (sometimes called consecutive interior angles). Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions Corresponding angles lie on the same side of the transversal and in corresponding positions. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions Alternate exterior angles are nonadjacent exterior angles that lie on opposite sides of the transversal. Copyright © 2014 Pearson Education, Inc.

Identifying Angle Pairs List all angle pairs in the figure. a. alternate interior b. corresponding Solution a. Copyright © 2014 Pearson Education, Inc.

Classifying an Angle Pair The photo shows the Hearst Building in New York City. The new tower (showing many triangles) was completed in 2006. Fill in the blank. a. 1 and 5 are _____________ angles. b. 2 and 7 are _____________ angles. alternate exterior same-side interior Copyright © 2014 Pearson Education, Inc.

Theorem 3.2 Alternate Interior Angles Theorem Theorem If two lines are cut by a transversal and a pair of alternate interior angles are congruent, then the two lines are parallel. If… Then… Copyright © 2014 Pearson Education, Inc.

Theorem 3.3 Corresponding Angles Theorem Theorem If two lines are cut by a transversal and a pair of corresponding angles are congruent, then the lines are parallel. If… Then… Copyright © 2014 Pearson Education, Inc.

Theorem 3.4 Alternate Exterior Angles Theorem Theorem If two lines are cut by a transversal and a pair of alternate exterior angles that are congruent, then the two lines are parallel. If… Then… Copyright © 2014 Pearson Education, Inc.

Theorem 3.5 Same-Side Interior Angles Theorem Theorem If two lines are cut by a transversal are two interior angles on the same side of the transversal are supplementary, then the two lines are parallel. If… Then… Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Theorem 3.6 Alternate Interior Angles Converse (Converse of Theorem 3.2) Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. If… Then… Copyright © 2014 Pearson Education, Inc.

Identifying Parallel Lines Which lines are parallel if Justify your answer. Solution are not formed by line l, so we concentrate on line a and line b with transversal m. are alternate interior angles. If , then a || b by the Alternate Interior Angles Theorem. Copyright © 2014 Pearson Education, Inc.

Determining Whether Lines Are Parallel The fence gate at the right is made up of pieces of wood arranged in various directions. Suppose Are lines r and s parallel? Explain. Solution Yes, r || s. are alternate exterior angles. If two lines and a transversal form congruent alternate exterior angles, then the lines are parallel (Alternate Exterior Angles Theorem). Copyright © 2014 Pearson Education, Inc.

Using Algebra to Prove Lines Are Parallel What is the value of x that makes a || b? Solution The two angles are same-side interior angles. By the Same-Side Interior Angles Theorem, a || b if the angles are supplementary. Copyright © 2014 Pearson Education, Inc.

Using Algebra to Prove Lines Are Parallel What is the value of x that makes a || b? (2x + 9) + 111 = 180 2x + 120 = 180 2x = 60 x = 30 Thus, if x = 30, then a || b. Copyright © 2014 Pearson Education, Inc.