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Preliminary concepts to understand
6.1 Relating Lines to Planes
Objective:
After studying this section, you will be able to understand basic concepts relating to planes, identify four methods of determining a plane, and apply two postulates concerning lines and planes.
A
B
m
Not a plane surface
B
C
Plane surface
Definition
Remember
If points, lines, segments, and so forth, lie in the same plane, we call
them coplanar. Points, lines, segments, and so forth that do not lie in
the same plane are called noncoplanar.
The point of intersection of a line and a plane
is called the foot of the line.
R
R
A
T
V
S
m
A
S
B
P
A, B, S, T, and V are coplanar points
AB and ST are coplanar lines
AB and ST are coplanar segments
A, B, S, T, and R are noncoplanar points
AB, ST, and RP are noncoplanar lines
AB, ST, and RP are noncoplanar
segments
T
V
m
B
P
Point V is the foot of the line RP in plane m.
Four ways to determine a plane
Postulate Three noncollinear points
determine a plane
A line and a point not on the line
determine a plane
k
B
A
Theorem
C
P
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Theorem
Two intersecting lines determine
a plane
Theorem
Two parallel lines determine a
plane
Two postulates concerning lines
and planes
Postulate If a line intersects a plane not
containing it, then the intersection
is exactly one point.
Postulate If two planes intersect, their
intersection is exactly one line.
A
m
k
B
k
B
A
m
U
Example 1
n=
A, B, and V determine plane _____
Name the foot of RS in m
AB and RS determine plane ____
n
R
W
B
V
P
A
Given: A, B, and C lie in plane m
PB ⊥ AB
P
PB ⊥ BC
AB ≅ BC
Prove ∠APB ≅ ∠CPB
B
m
A
C
m
S
AB and point ____ determine plane n
Does W lie in plane n?
Line AB and line ____ determine plane m
A, B, V, and ____ are coplanar points
A, B, V, and ____ are noncoplanar points
If R and S lie in plane n, what can be said
about RS?
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Definition
6.2 Perpendicularity of a Line and a Plane
A line is perpendicular to a plane if it is perpendicular
to every one of the lines in the plane that pass
through its foot.
A
Objective:
After studying this section, you will be able to recognize when a line is perpendicular to a plane and apply the basic theorem concerning the perpendicularity of a line and a plane.
E
m
C
B
D
We have two kinds of perpendicularity
Between two lines( AB
BD)
Between a line and a plane( AB
The definition is a powerful statement because of the
words every one. If we are given that AB
m (in the
diagram), we can draw three conclusions
A
m
C
B
E
D
AB
BC
AB
BD
AB
BE
Theorem
m)
If a line is perpendicular to two
distinct lines that lie in a plane
and that pass through its foot,
then it is perpendicular to the
plane.
A
C
F
m
B
Given: PF ⊥ k
Example
PG ≅ PH
If angle STR is a right angle, can we
conclude that line ST is perpendicular to m?
Why or why not?
P
Prove ∠G ≅ ∠H
F
k
G
H
S
T
m
R
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Given: B, C, D, and E lie in plane n
AB ⊥ n
HJJG
BE perpendicular bisector to CD
Prove: ΔADC is isosceles.
Summary
A
B
n
C
D
E
Make a diagram or use a physical model to
show how a line might be perpendicular to
one of two lines that passes through its foot
and not be perpendicular to the other one.
Homework: worksheet 6.2
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Definition
A line and a plane are parallel if they do not intersect.
6.3 Basic Facts About Parallel Planes
Objective:
After studying this section, you will be able to recognize lines parallel to planes, parallel planes, skew lines and use properties relating parallel lines and planes.
In the diagram below the planes are parallel but the
lines are not because A, B, C, D do not determine a
plane. These lines are called skew.
B
A
B
m
Two planes are parallel if they do not intersect
m
n
Theorem
If a plane intersects two parallel
planes, the lines of intersection
are parallel.
A
m
C
D
B
n
Definition
m
Two lines are skew if they are not
coplanar, are not parallel, and do not
intersect.
A
D
n
C
s
Properties Relating Parallel Lines and Planes
1. If two planes are perpendicular to the
same line, they are parallel to each other.
2. If a line is perpendicular to one of two
parallel planes, it is perpendicular to the other
plane as well.
3. If two planes are parallel to the same
plane, they are parallel to each other.
4. If two lines are perpendicular to the same
plane, they are parallel to each other.
5. If a plane is perpendicular to one of two
parallel lines, it is perpendicular to the other
line as well.
Given: m & n
HJJG
AB lies in m
HJJG
CD lies in n
HJJG HJJG
AC & BD
Prove: AD bisects BC
B
A
C
m
D
n
1