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Name_________________________________________________________ 3.2 Date __________ Parallel Lines and Transversals For use with Exploration 3.2 Essential Question When two parallel lines are cut by a transversal, which of the resulting pairs of angles are congruent? 1 EXPLORATION: Exploring Parallel Lines Go to BigIdeasMath.com for an interactive tool to investigate this exploration. 6 D 5 Work with a partner. Use dynamic geometry software to draw two parallel lines. Draw a third line that intersects both parallel lines. Find the measures of the eight angles that are formed. What can you conclude? 4 B 3 E 1 2 4 3 1 A 0 −3 2 F 6 8 7 5 2 −2 −1 0 1 2 3 4 C 5 6 EXPLORATION: Writing Conjectures Work with a partner. Use the results of Exploration 1 to write conjectures about the following pairs of angles formed by two parallel lines and a transversal. a. corresponding angles 1 2 4 3 5 6 8 7 Copyright © Big Ideas Learning, LLC All rights reserved. b. alternate interior angles 1 2 4 3 5 6 8 7 Geometry Student Journal 71 Name _________________________________________________________ Date _________ 3.2 2 Parallel Lines and Transversals (continued) EXPLORATION: Writing Conjectures (continued) c. alternate exterior angles 1 2 4 3 5 6 8 7 d. consecutive interior angles 1 2 4 3 5 6 8 7 Communicate Your Answer 3. When two parallel lines are cut by a transversal, which of the resulting pairs of angles are congruent? 4. In Exploration 2, m∠1 = 80°. Find the other angle measures. 72 Geometry Student Journal Copyright © Big Ideas Learning, LLC All rights reserved. Name_________________________________________________________ Date __________ Notetaking with Vocabulary 3.2 For use after Lesson 3.2 In your own words, write the meaning of each vocabulary term. corresponding angles parallel lines supplementary angles vertical angles Theorems Theorem 3.1 Corresponding Angles Theorem t If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Examples 1 2 3 4 In the diagram, ∠2 ≅ ∠6 and ∠3 ≅ ∠7. 5 6 7 8 Theorem 3.2 Alternate Interior Angles Theorem p q If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Examples In the diagram, ∠3 ≅ ∠6 and ∠4 ≅ ∠5. Theorem 3.3 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Examples In the diagram, ∠1 ≅ ∠8 and ∠2 ≅ ∠7. Copyright © Big Ideas Learning, LLC All rights reserved. Geometry Student Journal 73 Name _________________________________________________________ Date _________ 3.2 Notetaking with Vocabulary (continued) Theorem 3.4 Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. In the diagram, ∠3 and ∠5 are supplementary, and ∠ 4 and ∠6 are supplementary. Examples Notes: t 1 2 3 4 5 6 7 8 p q Extra Practice In Exercises 1–4, find m∠1 and m∠2. Tell which theorem you use in each case. 1. 110° 1 2 2. 63° 2 1 3. 1 95° 2 74 Geometry Student Journal Copyright © Big Ideas Learning, LLC All rights reserved. Name_________________________________________________________ 3.2 Date __________ Notetaking with Vocabulary (continued) 4. 1 2 101° In Exercises 5–8, find the value of x. Show your steps. 5. 110° (x + 12)° 6. (10x − 55)° 45° 7. 4x° 6 8. (2x − 3)° 52° 2 Copyright © Big Ideas Learning, LLC All rights reserved. 153° Geometry Student Journal 75