1. Lesson 10.4
Parallels in Space pp

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

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 C D C D n A B A B

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. n l m

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

22. p l q

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

24. p l q

25. q l p

26. p q l

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

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

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

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

42. ■ 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. A B C D E F G H

43. Analytic Geometry
Slopes of Parallel Lines

44. Slope measures the angle that a line makes with the horizontal axis.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.
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.
-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