Chapter 6 — Answers

Selected answers for Chapter 6: Parallel Lines, transcribed from the book’s appendix. They are meant for checking your work, not for copying — the book prints answers to selected exercises only.

Back to Chapter 6 · Teacher’s answers: Chapter 6 · Complete key: Answers to Exercises PDF · Printed answers: appendices PDF

Lesson 1 (pages 214–217)

4. Yes, because W and E are equidistant from the endpoints of NS.

12. 6 cm.

13. 90 ft.

15. In the west.

19. Seven.

21. Line l is the perpendicular bisector of ST.

22. If a line segment is bisected, it is divided into two equal segments.

31. 9.5 in.

34. C and E; D and F.

36. Because they are corresponding parts of congruent triangles (△ABC ≅ △ADC by SSS).

39. N.

43. Three.

Lesson 2 (pages 221–224)

1. (1) ∠1 = ∠2. (2) ∠2 = ∠3. (3) ∠1 = ∠3. (4) $a\parallel b$.

4. Alternate interior angles.

6. In a plane, two lines perpendicular to a third line are parallel.

7. Corresponding angles.

9. Equal corresponding angles mean that lines are parallel.

14. No.

18. $AE\parallel BF$.

27. No. These angles are not formed by a transversal. They could be equal and yet lines $x$ and $y$ could intersect either to the left or to the right.

32. Four.

35. SAS.

38. No.

Lesson 3 (pages 226–229)

1. 12.

2. If they form a right angle.

5. ∠3.

6. Equal corresponding angles mean that lines are parallel.

12. At least, no more than, exactly.

14. No more than.

20. They form equal corresponding angles with a transversal.

22. They seem to be supplementary.

25. All four of them.

27. Down, below the figure.

30. One.

33. In a plane, two lines perpendicular to a third line are parallel.

36. Through a point not on a line, there is exactly one line parallel to the line.

37. Yes. (SAS.)

42. $a$ and $b$ are not parallel.

43. Given.

Lesson 4 (pages 232–235)

1. They are equal.

2. Parallel lines form equal corresponding angles.

5. (1) Given. (2) Parallel lines form equal corresponding angles. (3) Vertical angles are equal. (4) Substitution.

9. Betweenness of Rays Theorem.

12. Equal corresponding angles mean that lines are parallel.

13. ∠1 = 115°, ∠2 = 65°, ∠3 = 115°.

17. They must be perpendicular because, in a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other.

20. 67°, 113°, 67°, and 113°.

22. ∠1 and ∠2 are supplementary because parallel lines form supplementary interior angles on the same side of a transversal.

24. ∠3 = ∠4 because parallel lines form equal alternate interior angles.

30. Through a point not on a line, there is exactly one line parallel to the line (the Parallel Postulate).

31. Parallel lines form equal alternate interior angles.

37. Two points determine a line.

39. In a plane, two lines perpendicular to a third line are parallel.

p. 752

44. The length of a perpendicular segment between them.

46. 70°.

Lesson 5 (pages 238–241)

1. The sum of the angles of a triangle is 180°.

2. Substitution.

3. Substitution.

4. Subtraction.

9. An equilateral triangle is equiangular.

10. An equiangular triangle has three equal angles.

14. Through a point not on a line, there is exactly one line parallel to the line.

16. Parallel lines form equal corresponding angles.

19. ∠2 > ∠1 and ∠2 > ∠A.

24. An equilateral triangle.

27. It is a right triangle.

28. They are complementary.

34. ∠2 and ∠3 are complementary.

36. Complements of the same angle are equal.

40. An exterior angle of a triangle is equal to the sum of the remote interior angles.

41. Substitution.

Lesson 6 (pages 245–248)

1. ASA.

5. HL.

6. Parallel lines form equal corresponding angles.

9. Three.

12. Yes.

19. HL.

21. If two angles of a triangle are equal, the sides opposite them are equal.

24. In a plane, two lines perpendicular to a third line are parallel.

27. Substitution.

29. AAS.

32. AB = AC, AD = AE, and ∠A = ∠A.

40. Because ∠DAP = ∠EAP (corresponding parts of congruent triangles are equal).

42. SSS.

44. 45°.

50. Substitution.

Review (pages 250–254)

1. Fold the figure along the proposed line of symmetry and see if the two halves coincide.

5. It is the perpendicular bisector of AB.

7. Equal corresponding angles mean that lines are parallel.

9. To the sum of the remote interior angles.

12. They are perpendicular.

14. Are equal, their sides are perpendicular.

16. In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment.

26. Equilateral.

30. In a plane, two lines perpendicular to a third line are parallel.

33. In a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other.

35. Concurrent.

39. ∠B = ∠CAD = 45°.

42. BA = CD by substitution.

48. ∠CDB = 2x°.

50. ∠ACB = 3x°.

Algebra Review (page 256)

1. $\dfrac{7}{6}$.

2. $\dfrac{x}{20}$.

3. $\dfrac{x+10}{2x}$.

4. $\dfrac{4x-3}{x^3}$.

5. $\dfrac{2}{3}$.

6. $\dfrac{x^6}{8}$.

7. $\dfrac{5}{7}$.

8. $\dfrac{x+2}{2}$.

9. $\dfrac{3x}{4}$.

10. $\dfrac{1}{5}$.

11. $\dfrac{1}{6}$.

12. $\dfrac{2}{x}$.

13. $\dfrac{4x+2y-z}{8}$.

14. $\dfrac{4\pi r^3}{3}$.

15. $\dfrac{x-y}{y}$.

16. $\dfrac{2(x-3)}{2x-3}$ or $\dfrac{2x-6}{2x-3}$.

17. $\dfrac{x+2}{x^2-1}$.

18. $\dfrac{1+x}{x-y}$.

19. 1.

20. $\dfrac{x^3}{x-1}$.