Chapter 1
Section 1.1
Identify Points, Lines, and Planes
Warm Up Problems
Solve the inequality. Graph the solution.
− 6 < x + 12
− 4 − 5 x ≤ 31
To define it or not to define it
• Definition: Using known words to define a
new word
• Undefined terms: When a term is not
given a definition
– Points, Lines, and Planes are all undefined
terms
– No formal definitions, but there is agreement
about what they mean
Undefined terms
• Point
– Something with no dimension
• Line
– Something that extends forever in one
dimension
• Plane
– Something that extends forever in two
dimensions
•
Points
• Points do not have actual size.
A
• How to Sketch:
Using dots
• How to label:
Use capital letters
Never name two points with the same letter
(in the same sketch).
B
A
C
Lines
• Lines extend indefinitely and have no thickness or width.
• How to sketch : using arrows at both ends.
n
A
B
C
• How to name: 2 ways
(1) small script letter – line n
     
(2) any two points on the line - AB , BC, AC , BA, CA, CB

• Never name a line using three points - ABC
Planes
• A plane is a flat surface that extends indefinitely in all directions.
• How to sketch: Use a parallelogram (four sided figure)
• How to name: 2 ways
(1) Capital script letter – Plane M
(2) Any 3 non collinear points in the plane - Plane: ABC/ ACB / BAC /
BCA / CAB / CBA
A
B
M
C
Horizontal Plane
Vertical Plane
Other
EXAMPLE 1
a.
Name points, lines, and planes
Give two other names for PQ and for plane R.
b. Name three points that are collinear.
Name four points that are coplanar.
SOLUTION
a.
Other names for PQ are QP and line n.
Other names for plane R are plane SVT
and plane PTV.
EXAMPLE 1
b.
Name points, lines, and planes
Points S, P, and T lie on the same line, so they
are collinear. Points S, P, T,and V lie in the same
plane, so they are coplanar.
GUIDED PRACTICE
1.
for Example 1
Use the diagram in Example 1. Give two other
names for ST . Name a point that is not coplanar
with points Q, S, and T.
ANSWER
TS, PT; point V
Defined Terms
• Collinear Points
– Points that lie on the same line
• Coplanar Points
– Points that lie in the same plane
• Intersect
– Two or more geometric figures are said to intersect if
they have one or more points in common
• Intersection
– Set of points the figures have in common
Defined Terms
• Line Segment
– A piece of a line cut off by two end points
• Ray
– All the points on one side of a given point
called the endpoint or initial point
End
Point
Symbolization
•
A
Lines
•
B
AB
Use two points on
the line and draw a
line above them
Rays
•
B
A
AB
Use the end point
and any other point
on the ray then
draw a ray on top
Opposite Rays: Two
rays that have the
same end point but
extend in opposite
directions
EXAMPLE 2
Name segments, rays, and opposite rays
a.
Give another name for GH .
b.
Name all rays with endpoint J .
Which of these rays are opposite rays?
SOLUTION
a.
Another name for GH is HG .
b.
The rays with endpoint J are JE , JG , JF , and JH .
The pairs of opposite rays with endpoint J are JE
and JF , and JG and JH .
GUIDED PRACTICE
2.
Give another name for EF
ANSWER
3.
for Example 2
FE
Are HJ and JH the same ray ? Are HJ and HG the
same ray? Explain.
ANSWER
No; the rays have different endpoints.
Yes; points J and G lie on the same side of H.
Intersection of Figures
The intersection of two figures is the set of points that are
common in both figures.
The intersection of two lines is a point.
m
Line m and line n intersect at point P.
P
n
Continued…….
3 Possibilities of Intersection of a Line and a Plane
(1) Line passes through plane – intersection is a point.
(2) Line lies on the plane - intersection is a line.
(3) Line is parallel to the plane - no common points.
Intersection of Two Planes is a
Line.
B
P
A
R

Plane P and Plane R intersect at the line AB
EXAMPLE 3
Sketch intersections of lines and planes
a.
Sketch a plane and a line that is in the plane.
b.
Sketch a plane and a line that does not intersect
the plane.
c.
Sketch a plane and a line that intersects the plane
at a point.
SOLUTION
a.
b.
c.
EXAMPLE 4
Sketch intersections of planes
Sketch two planes that intersect in a line.
SOLUTION
STEP 1
Draw: a vertical plane.
Shade the plane.
STEP 2
Draw: a second plane that is
horizontal. Shade this plane a different
color. Use dashed lines to show where
one plane is hidden.
STEP 3
Draw: the line of intersection.
GUIDED PRACTICE
4.
for Examples 3 and 4
Sketch two different lines that intersect a plane
at the same point.
ANSWER
Use the diagram at the right.
5.
Name the intersection of PQ and line k.
ANSWER
Point M
GUIDED PRACTICE
6.
Name the intersection of plane A and plane B.
ANSWER
7.
for Examples 3 and 4
Line k
Name the intersection of line k and plane A.
ANSWER
Line k
HW #1
p. 876: 1-6
p. 5-8: 1-16, 17-27
odd, 40-44, 47, 50,
53, 56