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Understanding Line Relationships: Parallel Lines and Angle Postulates

This lesson explores the connections between parallel lines and corresponding angles, enabling students to grasp the Converse of Corresponding Angles Postulate and the Parallel Postulate. It includes various examples illustrating how to identify and prove parallel lines through angle relationships, with practical applications and standardized test questions. The lesson incorporates a 5-Minute Check to reinforce learning on point-slope and slope-intercept forms. Students will enhance their understanding of geometric relationships and improve their problem-solving skills.

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Understanding Line Relationships: Parallel Lines and Angle Postulates

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 3–4) CCSS Then/Now Postulate 3.4: Converse of Corresponding Angles Postulate Postulate 3.5: Parallel Postulate Theorems: Proving Lines Parallel Example 1: Identify Parallel Lines Example 2: Standardized Test Example: Use Angle Relationships Example 3: Real-World Example: Prove Lines Parallel Lesson Menu

  3. containing the point (5, –2) in point-slope form? A. B. C. D. 5-Minute Check 1

  4. What is the equation of the line with slope 3 containing the point (–2, 7) in point-slope form? A.y = 3x + 7 B.y = 3x – 2 C.y – 7 = 3x + 2 D.y – 7 = 3(x + 2) 5-Minute Check 2

  5. What equation represents a line with slope –3 containing the point (0, 2.5) in slope-intercept form? A.y = –3x + 2.5 B.y = –3x C.y – 2.5 = –3x D.y = –3(x + 2.5) 5-Minute Check 3

  6. containing the point (4, –6) in slope-intercept form? A. B. C. D. 5-Minute Check 4

  7. What equation represents a line containing points (1, 5) and (3, 11)? A.y = 3x + 2 B.y = 3x – 2 C.y – 6 = 3(x – 2) D.y – 6 = 3x + 2 5-Minute Check 5

  8. A. B. C. D. 5-Minute Check 6

  9. Content Standards G.CO.9 Prove theorems about lines and angles. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others. CCSS

  10. You found slopes of lines and used them to identify parallel and perpendicular lines. • Recognize angle pairs that occur with parallel lines. • Prove that two lines are parallel. Then/Now

  11. Concept

  12. Concept

  13. Concept

  14. A. Given 1  3, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer. Identify Parallel Lines 1 and 3 are corresponding angles of lines a and b. Answer:Since 1  3, a║b by the Converse of the Corresponding Angles Postulate. Example 1

  15. B. Given m1 = 103 and m4 = 100, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer. Identify Parallel Lines 1 and 4 are alternate interior angles of lines a and c. Answer:Since 1 is not congruent to 4, line a is not parallel to line c by the Converse of the Alternate Interior Angles Theorem. Example 1

  16. A. Given 1  5, is it possible to prove that any of the lines shown are parallel? A. Yes; ℓ║ n B. Yes; m ║ n C. Yes; ℓ║ m D. It is not possible to prove any of the lines parallel. Example 1

  17. B. Given m4 = 105 and m5 = 70, is it possible to prove that any of the lines shown are parallel? A. Yes; ℓ║ n B. Yes; m║ n C. Yes; ℓ║ m D. It is not possible to prove any of the lines parallel. Example 1

  18. Find mZYN so that || . Show your work. Use Angle Relationships Read the Test Item From the figure, you know that mWXP = 11x – 25 and mZYN = 7x + 35. You are asked to find mZYN. Example 2

  19. m WXP = m ZYN Alternate exterior angles Use Angle Relationships Solve the Test ItemWXP and ZYN are alternate exterior angles. For line PQ to be parallel to line MN, the alternate exterior angles must be congruent. SomWXP = mZYN. Substitute the given angle measures into this equation and solve for x. Once you know the value of x, use substitution to findmZYN. 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. Example 2

  20. Since mWXP = mZYN,WXP ZYNand || . Use Angle Relationships Now use the value of x to findmZYN. mZYN = 7x + 35 Original equation = 7(15) + 35 x= 15 = 140 Simplify. Answer:mZYN = 140 Check Verify the angle measure by using the value of x to find mWXP. mWXP = 11x – 25 = 11(15) – 25 = 140 Example 2

  21. ALGEBRA Find x so that || . A.x = 60 B.x = 9 C.x = 12 D.x = 12 Example 2

  22. End of the Lesson

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