lesson 10.4 parallels in space pp. 428-431
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Lesson 10.4Parallels in Space
pp. 428-431
Lesson 10.4Parallels in Space
pp. 428-431
Objectives:1. To define parallel figures in space.2. To prove theorems about parallel
figures in space.
Objectives:1. To define parallel figures in space.2. To prove theorems about parallel
figures in space.
Parallel planesParallel planes are two planes are two planes that do not intersect.that do not intersect.
A A line parallel to a planeline parallel to a plane is a is a line that does not intersect the line that does not intersect the plane.plane.
DefinitionDefinitionDefinitionDefinition
Theorem 10.8Two lines perpendicular to the same plane are parallel.
Theorem 10.8Two lines perpendicular to the same plane are parallel.
mm
Theorem 10.9If two lines are parallel, then any plane containing exactly one of the two lines is parallel to the other line.
Theorem 10.9If two lines are parallel, then any plane containing exactly one of the two lines is parallel to the other line.
nn
CCDD mm
CCDD
AABB
AABB
Theorem 10.10A plane perpendicular to one of two parallel lines is perpendicular to the other line also.
Theorem 10.10A plane perpendicular to one of two parallel lines is perpendicular to the other line also.
nn
Theorem 10.11Two lines parallel to the same line are parallel.
Theorem 10.11Two lines parallel to the same line are parallel.
Theorem 10.12A plane intersects two parallel planes in parallel lines.
Theorem 10.12A plane intersects two parallel planes in parallel lines.
nn
mm
nnnn
mm
Theorem 10.13Two planes perpendicular to the same line are parallel.
Theorem 10.13Two planes perpendicular to the same line are parallel.
nn
mm
Theorem 10.14A line perpendicular to one of two parallel planes is perpendicular to the other also.
Theorem 10.14A line perpendicular to one of two parallel planes is perpendicular to the other also.
nn
mm
nn
mm
Theorem 10.15Two parallel planes are everywhere equidistant.
Theorem 10.15Two parallel planes are everywhere equidistant.
nn
mm
Two lines l and m are perpendicular to the same line but not parallel to each other. Name their relationship.
1. Parallel2. Skew3. Coplanar4. Perpendicular
Two lines l and m are perpendicular to the same line but not parallel to each other. Name their relationship.
1. Parallel2. Skew3. Coplanar4. Perpendicular
nn ll
mm
Given a line l and two planes p and q, suppose l || p. If l q, is p q?
1. Yes2. No
Given a line l and two planes p and q, suppose l || p. If l q, is p q?
1. Yes2. No
pp
llqq
Given a line l and two planes p and q, suppose l || p. If p q, is l q?
1. Yes2. No
Given a line l and two planes p and q, suppose l || p. If p q, is l q?
1. Yes2. No
pp
llqq
pp
ll
pp
qqll
Homeworkp. 431
Homeworkp. 431
►B. ExercisesDisprove each of these false statements by sketching a counterexample.
7. Two planes parallel to the same line are parallel.
►B. ExercisesDisprove each of these false statements by sketching a counterexample.
7. Two planes parallel to the same line are parallel.
►B. Exercises7.
►B. Exercises7.
►B. ExercisesDisprove each of these false statements by sketching a counterexample.
8. Two lines parallel to the same plane are parallel.
►B. ExercisesDisprove each of these false statements by sketching a counterexample.
8. Two lines parallel to the same plane are parallel.
►B. Exercises8.
►B. Exercises8.
►B. ExercisesDisprove 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. ExercisesDisprove 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. Exercises9.
►B. Exercises9.
►B. ExercisesDisprove 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. ExercisesDisprove 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. Exercises10.►B. Exercises10.
►B. ExercisesDisprove each of these false statements by sketching a counterexample.11. Lines perpendicular to parallel lines
are parallel.
►B. ExercisesDisprove each of these false statements by sketching a counterexample.11. Lines perpendicular to parallel lines
are parallel.
►B. Exercises11.►B. Exercises11.
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.19. Point G is interior to the prism.
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.19. Point G is interior to the prism.
AA
BB CC
DD
EE FF
GG
HH
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.20. DEF is a base of the prism.
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.20. DEF is a base of the prism.
AA
BB CC
DD
EE FF
GG
HH
AA
BB CC
DD
EE FF
GG
HH
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.
21. CD is an edge of the prism.
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.
21. CD is an edge of the prism.
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.22. DEF ABC
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.22. DEF ABC
AA
BB CC
DD
EE FF
GG
HH
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.23. If Q is between G and H, then
Q is interior to the prism.
■ Cumulative ReviewAnswer true or false. Refer to the prism shown.23. If Q is between G and H, then
Q is interior to the prism.AA
BB CC
DD
EE FF
GG
HH
Analytic Geometry
Slopes of Parallel Lines
Analytic Geometry
Slopes of Parallel Lines
Slope measures the angle that a line makes with the horizontal axis.
Slope measures the angle that a line makes with the horizontal axis.
1122
l1l1l2l2
Find the equation of the line through (-2, -1) and parallel to 3x + 4y = 2.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
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.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
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
Find the equation of the line through (3, -2) and parallel to 2x - y = 5.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
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.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
y - y1 = m(x - x1)
y - (-2) = 2(x - 3)
y + 2 = 2x - 6
y = 2x - 8