Geometry 201 unit 3.3
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This document discusses proving that two lines are parallel using theorems about angles formed when a transversal crosses the lines. It provides examples of using the converse of corresponding angles, alternate interior angles, same-side interior angles, and alternate exterior angles postulates/theorems to show that lines are parallel given information about the angles. It also includes examples applying these concepts to solve geometry problems and prove lines are parallel in practical applications like carpentry.
Geometry 201 unit 3.3
- 1. UNIT 3.3 PROVING LINESUNIT 3.3 PROVING LINES PARALLELPARALLEL
- 2. Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m∠A + m∠B = 90°, then ∠A and ∠B are complementary. 3. If AB + BC = AC, then A, B, and C are collinear. If a + c = b + c, then a = b. If ∠A and ∠ B are complementary, then m∠A + m∠B =90°. If A, B, and C are collinear, then AB + BC = AC.
- 3. Use the angles formed by a transversal to prove two lines are parallel. Objective
- 4. Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem.
- 6. Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. Example 1A: Using the Converse of the Corresponding Angles Postulate ∠4 ≅ ∠8 ∠4 ≅ ∠8 ∠4 and ∠8 are corresponding angles. ℓ || m Conv. of Corr. ∠s Post.
- 7. Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. Example 1B: Using the Converse of the Corresponding Angles Postulate m∠3 = (4x – 80)°, m∠7 = (3x – 50)°, x = 30 m∠3 = 4(30) – 80 = 40 Substitute 30 for x. m∠8 = 3(30) – 50 = 40 Substitute 30 for x. ℓ || m Conv. of Corr. ∠s Post. ∠3 ≅ ∠8 Def. of ≅ ∠s. m∠3 = m∠8 Trans. Prop. of Equality
- 8. Check It Out! Example 1a Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m∠1 = m∠3 ∠1 ≅ ∠3 ∠1 and ∠3 are corresponding angles. ℓ || m Conv. of Corr. ∠s Post.
- 9. Check It Out! Example 1b Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m∠7 = (4x + 25)°, m∠5 = (5x + 12)°, x = 13 m∠7 = 4(13) + 25 = 77 Substitute 13 for x. m∠5 = 5(13) + 12 = 77 Substitute 13 for x. ℓ || m Conv. of Corr. ∠s Post. ∠7 ≅ ∠5 Def. of ≅ ∠s. m∠7 = m∠5 Trans. Prop. of Equality
- 10. The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ.
- 12. Use the given information and the theorems you have learned to show that r || s. Example 2A: Determining Whether Lines are Parallel ∠4 ≅ ∠8 ∠4 ≅ ∠8 ∠4 and ∠8 are alternate exterior angles. r || s Conv. Of Alt. Int. ∠s Thm.
- 13. m∠2 = (10x + 8)°, m∠3 = (25x – 3)°, x = 5 Use the given information and the theorems you have learned to show that r || s. Example 2B: Determining Whether Lines are Parallel m∠2 = 10x + 8 = 10(5) + 8 = 58 Substitute 5 for x. m∠3 = 25x – 3 = 25(5) – 3 = 122 Substitute 5 for x.
- 14. m∠2 = (10x + 8)°, m∠3 = (25x – 3)°, x = 5 Use the given information and the theorems you have learned to show that r || s. Example 2B Continued r || s Conv. of Same-Side Int. ∠s Thm. m∠2 + m∠3 = 58° + 122° = 180° ∠2 and ∠3 are same-side interior angles.
- 15. Check It Out! Example 2a m∠4 = m∠8 Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. ∠4 ≅ ∠8 ∠4 and ∠8 are alternate exterior angles. r || s Conv. of Alt. Int. ∠s Thm. ∠4 ≅ ∠8 Congruent angles
- 16. Check It Out! Example 2b Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. m∠3 = 2x°, m∠7 = (x + 50)°, x = 50 m∠3 = 100° and m∠7 = 100° ∠3 ≅ ∠7 r||s Conv. of the Alt. Int. ∠s Thm. m∠3 = 2x = 2(50) = 100° Substitute 50 for x. m∠7 = x + 50 = 50 + 50 = 100° Substitute 5 for x.
- 17. Example 3: Proving Lines Parallel Given: p || r , ∠1 ≅ ∠3 Prove: ℓ || m
- 18. Example 3 Continued Statements Reasons 1. p || r 5. ℓ ||m 2. ∠3 ≅ ∠2 3. ∠1 ≅ ∠3 4. ∠1 ≅ ∠2 2. Alt. Ext. ∠s Thm. 1. Given 3. Given 4. Trans. Prop. of ≅ 5. Conv. of Corr. ∠s Post.
- 19. Check It Out! Example 3 Given: ∠1 ≅ ∠4, ∠3 and ∠4 are supplementary. Prove: ℓ || m
- 20. Check It Out! Example 3 Continued Statements Reasons 1. ∠1 ≅ ∠4 1. Given 2. m∠1 = m∠4 2. Def. ≅ ∠s 3. ∠3 and ∠4 are supp. 3. Given 4. m∠3 + m∠4 = 180° 4. Trans. Prop. of ≅ 5. m∠3 + m∠1 = 180° 5. Substitution 6. m∠2 = m∠3 6. Vert.∠s Thm. 7. m∠2 + m∠1 = 180° 7. Substitution 8. ℓ || m 8. Conv. of Same-Side Interior ∠s Post.
- 21. Example 4: Carpentry Application A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m∠1= (8x + 20)° and m∠2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.
- 22. Example 4 Continued A line through the center of the horizontal piece forms a transversal to pieces A and B. ∠1 and ∠2 are same-side interior angles. If ∠1 and ∠2 are supplementary, then pieces A and B are parallel. Substitute 15 for x in each expression.
- 23. Example 4 Continued m∠1 = 8x + 20 = 8(15) + 20 = 140 m∠2 = 2x + 10 = 2(15) + 10 = 40 m∠1+m∠2 = 140 + 40 = 180 Substitute 15 for x. Substitute 15 for x. ∠1 and ∠2 are supplementary. The same-side interior angles are supplementary, so pieces A and B are parallel by the Converse of the Same-Side Interior Angles Theorem.
- 24. Check It Out! Example 4 What if…? Suppose the corresponding angles on the opposite side of the boat measure (4y – 2)° and (3y + 6)°, where y = 8. Show that the oars are parallel. 4y – 2 = 4(8) – 2 = 30° 3y + 6 = 3(8) + 6 = 30° The angles are congruent, so the oars are || by the Conv. of the Corr. ∠s Post.
- 25. Lesson Quiz: Part I Name the postulate or theorem that proves p || r. 1. ∠4 ≅ ∠5 Conv. of Alt. Int. ∠s Thm. 2. ∠2 ≅ ∠7 Conv. of Alt. Ext. ∠s Thm. 3. ∠3 ≅ ∠7 Conv. of Corr. ∠s Post. 4. ∠3 and ∠5 are supplementary. Conv. of Same-Side Int. ∠s Thm.
- 26. Lesson Quiz: Part II Use the theorems and given information to prove p || r. 5. m∠2 = (5x + 20)°, m ∠7 = (7x + 8)°, and x = 6 m∠2 = 5(6) + 20 = 50° m∠7 = 7(6) + 8 = 50° m∠2 = m∠7, so ∠2 ≅ ∠7 p || r by the Conv. of Alt. Ext. ∠s Thm.
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