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Chapter 3-5

Chapter 3-5. Proving Lines Parallel. Recognize angle conditions that occur with parallel lines. Prove that two lines are parallel based on given angle relationships.

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Chapter 3-5

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  1. Chapter 3-5 Proving Lines Parallel

  2. Recognize angle conditions that occur with parallel lines. • Prove that two lines are parallel based on given angle relationships. Standard 7.0Students prove and use theorems involving the properties of parallel lines cut by a transversal,the properties of quadrilaterals, and the properties of circles. (Key) Standard 16.0Students perform basic constructions with a straightedge and compass,such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.(Key) Lesson 3-5 Ideas/Vocabulary

  3. p q r • If two lines are parallel to the same line, then they are parallel to each other. • If p // q and q // r, then p // r. Transitive property of Parallels

  4. Reminders from Section 1 We will use these same theorems to prove the lines are parallel given certain angle information.

  5. Corresponding Angle Theorem If two parallel lines are cut by a transversal, then corresponding angles are congruent. // lines  corresponding s are 

  6. Corresponding Angle Theorem

  7. Alternate Interior Angle Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. // lines  Alt. Int. s are 

  8. Alternate Interior Angle Theorem

  9. Alternate Exterior Angle Theorem If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. // lines  Alt. Ext. s are 

  10. Alternate Exterior Angle Theorem

  11. Consecutive Interior Angle Theorem If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. // lines  Consec. Int. s are Supp.

  12. Consecutive Interior Angle Theorem 1 2 m1 + m2 = 180

  13. p m n Two  Theorem • If two lines are perpendicular to the same line, then they are parallel to each other. • If m  p and n  p, then m // n.

  14. Animation: Construct a Parallel Line Through a Point not on Line Lesson 3-5 Postulates

  15. Lesson 3-5 Theorems

  16. Identify Parallel Lines Determine which lines, if any, are parallel. Consec. Int. s are supp. 77o  a//b Alt. Int. s are not   a is not // c Consec. Int. s are not supp.  b is not // c Lesson 3-5 Example 1

  17. A • B • C • D Determine which lines, if any are parallel.I. e || fII. e || gIII. f || g I only II only III only I, II, and III Lesson 3-5 CYP 1

  18. ALGEBRA Find x and m ZYN so that || . ExploreFrom the figure, you know that m WXP = 11x – 25 and m ZYN = 7x + 35. You also know that WXP and ZYN are alternate exterior angles. Solve Problems with Parallel Lines Lesson 3-5 Example 2

  19. ALGEBRA Find x and m ZYN so that || . m WXP = m ZYN Alternate exterior  thm. If Alt. Ext. angles are , then the lines will be // 11x – 25 = 7x + 35 Substitution 4x – 25 = 35 Subtract 7x from each side. 4x = 60 Add 25 to each side. x = 15 Divide each side by 4. Lesson 3-5 Example 2

  20. Now use the value of x to findm ZYN. m ZYN = 7x + 35 Original equation Answer:x = 15, m ZYN = 140 Solve Problems with Parallel Lines = 7(15) + 35 x= 15 = 140 Simplify. Lesson 3-5 Example 2

  21. ALGEBRA Find x so that || . x = 60 x = 9 x = 12 x = 12 • A • B • C • D Lesson 3-5 CYP 2

  22. Given: ℓ || m Prove: r || s Prove Lines Parallel Lesson 3-5 Example 3

  23. Proof: Statements Reasons 1.1. Given 2.2. Consecutive Interior Angle Theorem 3.3. Definition of supplementary angles 4.4. Definition of congruent angles 5.5. Substitution 6.6. Definition of supplementary angles 7. 7. If consecutive interior angles theorem Prove Lines Parallel Lesson 3-5 Example 3

  24. A • B • C Given x || y and , can you use theCorresponding Angles Postulate to prove a || b? yes no not enough informationto determine Lesson 3-5 CYP 3

  25. Slope and Parallel Lines Determine whether p || q. slope of p: slope of q: Answer: Since the slopes are equal, p || q. Lesson 3-5 Example 4

  26. Determine whether r || s. Yes, ris parallel to s. No, r is not parallel to s. It cannot be determined. • A • B • C Lesson 3-5 CYP 4

  27. Homework Chapter 3-5 • Pg 175 1 – 5, 7 – 19, 23 (proof), 24(proof), 37, 50 – 52

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