1. 3.2 Properties of Parallel Lines Ms. Kelly Fall 2010

2. Standards/Objectives: Objectives: State and apply a postulate or theorems about parallel lines

3. Postulate 10 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent. 1 2  1 ≅  2

4. Theorem 3.2 Alternate Interior Angles If two parallel lines are cut by a transversal, then alternate interior angles are congruent. 3 4  3 ≅  4

5. Theorem 3.3 Same-Side Interior Angles If two parallel lines are cut by a transversal, then same-side interior angles are supplementary. 5 6  5 +  6 = 180 °

6. Alternate Exterior Angles If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. 7 8  7 ≅  8

7. Theorem 3.4 Perpendicular Transversal If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other. j  k j h k

8. Example 1: Proving the Alternate Interior Angles Theorem Given: p ║ q Prove:  1 ≅  2 1 2 3

9. Proof Statements: 1.p ║ q 2.  1 ≅  3 3.  3 ≅  2 4.  1 ≅  2 Reasons: 1.Given 2.Corresponding Angles Postulate 3.Vertical Angles Theorem 4.Transitive Property of Congruence

10. Example 2: Using properties of parallel lines Given that m  5 = 65 °, find each measure. A. m  6B. m  7 C. m  8D. m  9 6 7 5 8 9

11. Solutions: a.m  6 = m  5 = 65 ° b.m  7 = 180 ° - m  5 =115 ° c.m  8 = m  5 = 65 ° d.m  9 = m  7 = 115 °

12. Ex. 3—Classifying Leaves BOTANY—Some plants are classified by the arrangement of the veins in their leaves. In the diagram below, j ║ k. What is m  1? 120 ° j k 1

13. Solution 1.m  1 + 120 ° = 180° 2.m  1 = 60 ° 1.Consecutive Interior angles Theorem 2.Subtraction POE

14. Ex. 4: Using properties of parallel lines Use the properties of parallel lines to find the value of x. 125 ° 4 (x + 15) °

15. 3.2 Day 2 In the four squares below, 4 of the 5 theorems/postulates will be used heavily for proofs Postulate 10Theorem 3-2 Theorem 3-3Theorem 3-4 1 2 2 1 1 2 2 1

16. Let’s review Example 1: Theorem 3-2 Proving the Alternate Interior Angles Theorem Given: p ║ q Prove:  1 ≅  2 1 2 3

17. Proof Statements: 1.p ║ q 2.  1 ≅  3 3.  3 ≅  2 4.  1 ≅  2 Reasons: 1.Given 2.Corresponding Angles Postulate (Postulate 10) 3.Vertical Angles Theorem (Theorem 2-3) 4.Transitive Property of Congruence

18. You try (we try): Given: K || n; transversal t cuts k and n. Prove: <1 is supplementary to <4 1 42

19. Solution Let’s use what we know about our theorems StatementsReasons 1.k || n; transversal t cuts k and n1. Given 2.  1 ≅  22. Theorem 3-2 (alt. int. angles) 3.  4 +  2 = 1803. Angle Addition Postulate 4.  4 +  1 = 1804. Substitution Prop 5.  4 is supplementary to  1 4. Def. of supplementary angles

20. Open your book to page 80 Complete 2 through 9 Your word bank: –Post 10 –Thm 3-2 –Thm 3-3 –Thm 3-4 –Vertical Angles thm

21. Complete on your own #20 and #21 on page 82 Ask yourself the following questions: –What am I proving (what kind of angles are they)? –How do I get there using the other theorems and postulates?

22. Now onto algebraic examples!!!!! Review of page 80 10-13 10.Angles 4, 5, 8 = 130; angles 2, 3, 6, 7 = 50 11.Angles 4, 5, 8 = x; angles 2, 3, 6, 7 = 180-x 12. 60 13. 100 In the next few examples, the markings are the most important thing when it comes to finding the angle values!

23. Algebraic Example 1

24. Algebraic Example 2

25. Algebraic Example 3

26. Algebraic Problems – you try

27. Closure – will be collected and graded On a small piece of paper, answer the following: 1.What theorem discusses same-side interior angles that are supplementary? 2.Postulate 10 discusses……. 3.What theorem discusses alternate interior angles? 4.Solve:

28. Groupwork Please complete the worksheet in your group and hand in for a grade. Then you may start your homework.

29. Homework Page 81 8-12, 15, 16