Lesson 10.4 Parallels in Space pp. 428-431

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

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.

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

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

m C D C D n A B A B

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

n

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

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

m n

m n n

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

m n

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

m m n n

Theorem 10.15 Two parallel planes are everywhere equidistant.

m n

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

n l m

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

p l q

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

p l q

q l p

p q l

Homework p. 431

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

►B. Exercises 7.

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

►B. Exercises 8.

►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.

►B. Exercises 9.

►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.

►B. Exercises 10.

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

►B. Exercises 11.

■ 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

■ 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

■ 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

■ 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

■ 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

Analytic Geometry Slopes of Parallel Lines

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

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

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

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

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