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Points, Lines, & Planes
Objectives
• Identify and correctly label points, lines, line segments,
rays, and planes.
pointpointpointpoint – a location in space, represented by a capital letter.
(No dimension)
• A Point A
Notation: A point is always labeled with a capital printed
letter. It has no size – one point is not bigger than
another.another.
linelinelineline – a straight path that has no thickness and extends
forever in both directions. (1 dimension)
Example:
•
•
A
B m
AB or BA or line m
Notation: A line is labeled either by two points or a single
lowercase letter. The order of the two letters does not
matter. (AB is the same as BA.)
collinearcollinearcollinearcollinear – points that lie on the same line.
Example:
Points A, B, and C are collinear.
•
•
A
C
B
•
D
•
noncollinearnoncollinearnoncollinearnoncollinear – points that do not lie on the same line.
Points A, B, and D are noncollinear.
Two points are always collinear.
(note)
Examples
1. Are points G, H, and J collinear or
noncollinear?
2. Are points F, H, and K collinear or
F
•
H
•
G
•
J
•
K
•
2. Are points F, H, and K collinear or
noncollinear?
3. Are points J and K collinear or
noncollinear?
Examples
1. Are points G, H, and J collinear or
noncollinear?
2. Are points F, H, and K collinear or
F
•
H
•
G
•
J
•
K
•
collinear
2. Are points F, H, and K collinear or
noncollinear?
3. Are points J and K collinear or
noncollinear?
noncollinear
collinear
line segment – part of a line consisting of two points, called
endpointsendpointsendpointsendpoints, and all points between them.
ray – part of a line that starts at an endpoint and extends
forever in one direction.
A
B
orAB BA
Notation: The order of the letters does matter for the
name of a ray.
A
B
•
AB
AB is not the same as BA.
opposite rays – two rays that have a common endpoint and
form a line.
Q
• R
• S
•
and are opposite raysRS RQ
You could also write RQ as QR (note the arrow
direction), although it’s a little confusing to read.
planeplaneplaneplane – a flat surface that has no thickness and extends
forever. (2 dimensions)
A•
B•
C•
R
Plane ABC or plane R
Notation: A plane is named either by a script capital letter
or three noncollinear points.
coplanarcoplanarcoplanarcoplanar – points that lie in the same plane.
Example:
Points A, B, C, and D are coplanar.
A
•
B
•
C
•
R
E
•
D
•
Points A, B, C, and D are coplanar.
noncoplanarnoncoplanarnoncoplanarnoncoplanar – points that do not lie in the same plane.
Points A, B, C, and E are noncoplanar.
Three noncollinear points are always coplanar.
Example Find three different ways you can name the
plane below.
S
F R
D
E

 


E
T 
Example Find three different ways you can name the
plane below.
S
F R
D
E

 


PlanePlanePlanePlane TTTT or any three letters exceptor any three letters exceptor any three letters exceptor any three letters except SSSS
E
T 
undefined termundefined termundefined termundefined term – a basic figure that cannot be defined in
terms of other figures
Points, lines, and planes are undefined terms – all other
geometric figures are defined in terms of them. Our
“definitions” are just descriptions of their attributes.
postulatepostulatepostulatepostulate – a statement that is accepted as true withoutpostulatepostulatepostulatepostulate – a statement that is accepted as true without
proof. Also called an axiom.
Postulate: Through any two points there is exactly one line.
(Euclid’s first postulate)
Postulate: Through any three noncollinear points there is
exactly one plane containing them.
Postulate: If two lines intersect, then they intersect in
exactly one point.exactly one point.

Postulate: If two planes intersect, then they intersect in
exactly one line.
Examples:
1. The intersection of planes H
and E is ____ or ____.
2. The intersection of m and n
is ____.
3. Line k intersects E at ____.
k
E
H
m
n
R
X
S
Y


4. R, X, and S are __________.
n
Examples:
1. The intersection of planes H
and E is ____ or ____.
2. The intersection of m and n
is ____.
3. Line k intersects E at ____.
n
X
Y
k
E
H
m
n
R
X
S
Y


RS
4. R, X, and S are __________.collinear
n

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2.1 Points, Lines, and Planes

  • 1. Points, Lines, & Planes Objectives • Identify and correctly label points, lines, line segments, rays, and planes.
  • 2. pointpointpointpoint – a location in space, represented by a capital letter. (No dimension) • A Point A Notation: A point is always labeled with a capital printed letter. It has no size – one point is not bigger than another.another.
  • 3. linelinelineline – a straight path that has no thickness and extends forever in both directions. (1 dimension) Example: • • A B m AB or BA or line m Notation: A line is labeled either by two points or a single lowercase letter. The order of the two letters does not matter. (AB is the same as BA.)
  • 4. collinearcollinearcollinearcollinear – points that lie on the same line. Example: Points A, B, and C are collinear. • • A C B • D • noncollinearnoncollinearnoncollinearnoncollinear – points that do not lie on the same line. Points A, B, and D are noncollinear. Two points are always collinear. (note)
  • 5. Examples 1. Are points G, H, and J collinear or noncollinear? 2. Are points F, H, and K collinear or F • H • G • J • K • 2. Are points F, H, and K collinear or noncollinear? 3. Are points J and K collinear or noncollinear?
  • 6. Examples 1. Are points G, H, and J collinear or noncollinear? 2. Are points F, H, and K collinear or F • H • G • J • K • collinear 2. Are points F, H, and K collinear or noncollinear? 3. Are points J and K collinear or noncollinear? noncollinear collinear
  • 7. line segment – part of a line consisting of two points, called endpointsendpointsendpointsendpoints, and all points between them. ray – part of a line that starts at an endpoint and extends forever in one direction. A B orAB BA Notation: The order of the letters does matter for the name of a ray. A B • AB AB is not the same as BA.
  • 8. opposite rays – two rays that have a common endpoint and form a line. Q • R • S • and are opposite raysRS RQ You could also write RQ as QR (note the arrow direction), although it’s a little confusing to read.
  • 9. planeplaneplaneplane – a flat surface that has no thickness and extends forever. (2 dimensions) A• B• C• R Plane ABC or plane R Notation: A plane is named either by a script capital letter or three noncollinear points.
  • 10. coplanarcoplanarcoplanarcoplanar – points that lie in the same plane. Example: Points A, B, C, and D are coplanar. A • B • C • R E • D • Points A, B, C, and D are coplanar. noncoplanarnoncoplanarnoncoplanarnoncoplanar – points that do not lie in the same plane. Points A, B, C, and E are noncoplanar. Three noncollinear points are always coplanar.
  • 11. Example Find three different ways you can name the plane below. S F R D E      E T 
  • 12. Example Find three different ways you can name the plane below. S F R D E      PlanePlanePlanePlane TTTT or any three letters exceptor any three letters exceptor any three letters exceptor any three letters except SSSS E T 
  • 13. undefined termundefined termundefined termundefined term – a basic figure that cannot be defined in terms of other figures Points, lines, and planes are undefined terms – all other geometric figures are defined in terms of them. Our “definitions” are just descriptions of their attributes. postulatepostulatepostulatepostulate – a statement that is accepted as true withoutpostulatepostulatepostulatepostulate – a statement that is accepted as true without proof. Also called an axiom.
  • 14. Postulate: Through any two points there is exactly one line. (Euclid’s first postulate) Postulate: Through any three noncollinear points there is exactly one plane containing them. Postulate: If two lines intersect, then they intersect in exactly one point.exactly one point. 
  • 15. Postulate: If two planes intersect, then they intersect in exactly one line.
  • 16. Examples: 1. The intersection of planes H and E is ____ or ____. 2. The intersection of m and n is ____. 3. Line k intersects E at ____. k E H m n R X S Y   4. R, X, and S are __________. n
  • 17. Examples: 1. The intersection of planes H and E is ____ or ____. 2. The intersection of m and n is ____. 3. Line k intersects E at ____. n X Y k E H m n R X S Y   RS 4. R, X, and S are __________.collinear n