Lesson 10.4 Parallels in Space pp. 428-431

1. Lesson 10.4 Parallels in Space pp. 428-431

2. Objectives: 1. To define parallel figures in space. 2. To prove theorems about parallel figures in space.

3. Definition Parallel planes are two planes that do not intersect. A line parallel to a plane is a line that does not intersect the plane.

4. m Theorem 10.8 Two lines perpendicular to the same plane are parallel.

5. Theorem 10.9 If two lines are parallel, then any plane containing exactly one of the two lines is parallel to the other line.

6. m D D C C B B A A n

7. Theorem 10.10 A plane perpendicular to one of two parallel lines is perpendicular to the other line also.

8. n

9. Theorem 10.11 Two lines parallel to the same line are parallel.

10. Theorem 10.12 A plane intersects two parallel planes in parallel lines.

11. m n

12. m n n

13. Theorem 10.13 Two planes perpendicular to the same line are parallel.

14. m n

15. Theorem 10.14 A line perpendicular to one of two parallel planes is perpendicular to the other also.

16. m m n n

17. Theorem 10.15 Two parallel planes are everywhere equidistant.

18. m n

19. Two lines l and m are perpendicular to the same line but not parallel to each other. Name their relationship. 1. Parallel 2. Skew 3. Coplanar 4. Perpendicular

20. m l n

21. Given a line l and two planes p and q, suppose l || p. If l q, is p  q? 1. Yes 2. No

22. q l p

23. Given a line l and two planes p and q, suppose l || p. If p  q, is l q? 1. Yes 2. No

24. q l p

25. q l p

26. l q p

27. Homework p. 431

28. ►B. Exercises Disprove each of these false statements by sketching a counterexample. 7. Two planes parallel to the same line are parallel.

29. ►B. Exercises 7.

30. ►B. Exercises Disprove each of these false statements by sketching a counterexample. 8. Two lines parallel to the same plane are parallel.

31. ►B. Exercises 8.

32. ►B. Exercises Disprove each of these false statements by sketching a counterexample. 9. If two planes are parallel, then any line in the first plane is parallel to any line in the second.

33. ►B. Exercises 9.

34. ►B. Exercises Disprove each of these false statements by sketching a counterexample. 10. If a line is parallel to a plane, then the line is parallel to every line in the plane.

35. ►B. Exercises 10.

36. ►B. Exercises Disprove each of these false statements by sketching a counterexample. 11. Lines perpendicular to parallel lines are parallel.

37. ►B. Exercises 11.

38. A B C G D E F H ■ Cumulative Review Answer true or false. Refer to the prism shown. 19. Point G is interior to the prism.

39. A B C G D E F H ■ Cumulative Review Answer true or false. Refer to the prism shown. 20. DEF is a base of the prism.

40. ■ Cumulative Review Answer true or false. Refer to the prism shown. 21. CD is an edge of the prism. A B C G D E F H

41. A B C G D E F H ■ Cumulative Review Answer true or false. Refer to the prism shown. 22. DEF  ABC

42. A B C G D E F H ■ Cumulative Review Answer true or false. Refer to the prism shown. 23. If Q is between G and H, then Q is interior to the prism.

43. Analytic Geometry Slopes of Parallel Lines

44. Slope measures the angle that a line makes with the horizontal axis. l2 l1 2 1

45. Find the equation of the line through (-2, -1) and parallel to 3x + 4y = 2. 1. Find the slope. 4y = -3x + 2 y = -3/4x + 1/2 m = -3/4

46. Find the equation of the line through (-2, -1) and parallel to 3x + 4y = 2. 2. Find the equation. y - y1 = m(x - x1) y - (-1) = -3/4(x - (-2)) y + 1 = -3/4x - 3/2 y = -3/4x - 5/2

47. Find the equation of the line through (3, -2) and parallel to 2x - y = 5. 2x - y = 5 -y = -2x + 5 y = 2x - 5 m = 2

48. Find the equation of the line through (3, -2) and parallel to 2x - y = 5. y - y1 = m(x - x1) y - (-2) = 2(x - 3) y + 2 = 2x - 6 y = 2x - 8