1. Section 3.2: Angles and Parallel Lines By Ryan Siebecker

2. Objectives 1.Use theorems to determine relationships between different pairs of angles 2.Use algebra to find angle measures

3. What are we doing? In this lesson, we will learn the relationship between angles formed by parallel lines and a transversal. Later, we will use these ideas to find solutions to algebraic problems. Line Transversal Tick marks show parallel lines

4. Congruent Angles Formed by ∥ Lines Alternate Interior ∠ ’s Alternate Exterior ∠ ’s Corresponding ∠ ’s 12 3 4 56 78 ∠ 3 ≅∠ 6 ∠ 5 ≅∠ 4 ∠ 1 ≅∠ 8 ∠ 7 ≅∠ 2 ∠ 7 ≅∠ 3 ∠ 8 ≅∠ 4 ∠ 1 ≅∠ 5 ∠ 2 ≅∠ 6

6. Congruent Angles: Shortcuts Alternate Interior Angles If angles are traced, they will form a “Z” or an “N” Ex. ∠ 3 and ∠ 6 are Alt. Int. ∠ ’s 12 34 56 78

7. Congruent Angles: Shortcuts Corresponding Angles If angles are traced, they will form a “F”. Can be backwards, upside down, or both. Ex. ∠ 2 and ∠ 6 are Corresponding ∠ ’s 12 3 4 56 78

8. Congruent Angles: Shortcuts Alternate Exterior Angles If angles are traced, they will form a “><” or similar pattern. Ex. ∠ 1 and ∠ 8 are Alt. Ext. ∠ ’s 12 34 56 78

9. Supplementary Angles Formed by ∥ Lines Same Side Interior ∠ ’s Same Side Exterior ∠ ’s ∠ 2 suppl. ∠ 8 ∠ 7 suppl. ∠ 1 Quick Question: What other angles in the diagram are supplementary? These angles are found using previously learned methods. (Hint: 8 pairs) 12 3 4 56 78 ∠ 3 suppl. ∠ 5 ∠ 4 suppl. ∠ 6 ∠ 1 suppl. ∠ 2 ∠ 1 suppl. ∠ 3 ∠ 2 suppl. ∠ 4 ∠ 4 suppl. ∠ 3 ∠ 5 suppl. ∠ 6 ∠ 5 suppl. ∠ 7 ∠ 6 suppl. ∠ 8 ∠ 7 suppl. ∠ 8 **Any angle pair that is not deemed as congruent (alt. int. ∠ ’s, alt. ext. ∠ ’s, corres. ∠ ’s) is supplementary. ”Same Side” comes from the fact that theses angles are on the same side of the transversal…

11. Sample Problem: Congruent Angles Name the ≅ ∠ ’s in the diagram formed by methods learned today: 1. Check for Alt. Ext. ∠ ’s (“><“) ∠ 1, ∠ 8 ∠ 2, ∠ 7 2. Check for Alt. Int. ∠ ’s (“Z” or “N”) ∠ 3, ∠ 6 ∠ 5, ∠ 4 3. Check for corresponding ∠ ’s (“F”) ∠ 8, ∠ 4 ∠ 7, ∠ 3 ∠ 5, ∠ 1 ∠ 2, ∠ 6 12 34 56 78

12. Sample Problem: Supplementary Angles Name the suppl. ∠ ’s in the diagram formed via methods learned today: 1. Check for Same Side Int. ∠ ’s ∠ 3, ∠ 5 ∠ 4, ∠ 6 2. Check for Same Side Ext. ∠ ’s ∠ 1, ∠ 7 ∠ 2, ∠ 8 12 34 56 78

13. Suppl. and Congruent ∠ ’s in Algebra 12 3 4 56 78

16. Congruent ∠ ’s in Algebra Sample Problem 12 3 4 56 78

17. Suppl. ∠ ’s in Algebra Sample Problem 12 3 4 56 78

18. Perpendicular Transversals… 1 2 43 5 6 7 8

19. Theorems The concepts learned today all have their respective theorems. Use these theorems in proofs to illustrate geometric ideas. Theorem 1: Corresponding Angles If two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent. Shortcut: ∥ lines => ≅ corres. ∠ ’s Theorem 2: Alternate Interior Angles If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent. Shortcut: ∥ lines => ≅ alt. int. ∠ ’s Theorem 3: Same Side Interior Angles If two parallel lines are cut by a transversal, then each pair of same side interior angles is supplementary. Shortcut: ∥ lines => suppl. same side int. ∠ ’s Theorem 4: Alternate Exterior Angles If two parallel lines are cut by a transversal, then each pari of alternate exterior angles is congruent. Shortcut: ∥ lines => ≅ alt. ext. ∠ ’s Theorem 5: Perpendicular Transversal In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

20. Practice Problems: You Try It! Name every pair of ≅ ∠ ’s in the diagram shown that are formed using today’s learned methods Alt. Ext. Angles ∠ 1, ∠ 8 ∠ 7, ∠ 2 Alt. Int. Angles ∠ 3, ∠ 6 ∠ 5, ∠ 4 Corresponding ∠ ’s ∠ 8, ∠ 4 ∠ 7, ∠ 3 ∠ 5, ∠ 1 ∠ 2, ∠ 6 12 3 4 5 6 78

21. Practice Problems: You Try It! Name every pair of supplementary ∠ ’s in the diagram shown, that are formed using methods learned today Same Side Interior ∠ 3, ∠ 5 ∠ 4, ∠ 6 Same Side Exterior ∠ 1, ∠ 7 ∠ 2, ∠ 8 12 3 4 5 6 78

22. Practice Problems: You Try It! 12 3 4 56 78

23. 12 3 4 56 78

24. Complete the two column proof 12 3 4 56 78 Statements Reasons 1.Given 2.Given 3.Def. of Congruent ∠ ’s 4. ≅ corres. ∠ ’s => ∥ lines

25. Works Cited Carter, John A. "Angles and Parallel Lines." Geometry. Columbus, OH: Glencoe/McGraw-Hill Education, 2012. 180-83. Print. Roberts, Donna. "Practice with Parallel Angles." Practice with Parallel Angles. Oswego City School District Regents Exam Prep Center, 1998. Web. 14 Jan. 2016.