Math 7 geometry 01 undefined terms rev 2

A Mathematics 7 Lecture
The Basics of Geometry

QUESTION!
What words come to
mind when you hear
the word
GEOMETRY?
The Basics of Geometry for High School

Geometry
Geometry comes from
the Greek words geo
(earth) and metron
(measure), so geometry
literally means “earth
measure”
Geometry is the branch of mathematics
that defines and relates the basic
properties and measurements of shapes

Geometry
The ancient Egyptians, Chinese,
Babylonians, Romans, and Greeks used
geometry for surveying, navigation,
astronomy, and other practical
occupations. The Greeks sought to
systematize the geometric facts they
knew by establishing logical reasons for
them and relationships among them.

The systematizing of
geometric facts and
principles culminated in the
geometry text Elements,
written in approximately 325
B.C. by Euclid. This most
remarkable text has been in
use for over 2000 years.
This recognizes Euclid as the
Father of Geometry.
Euclid
Father of Geometry
Geometry

Geometry in Art and Nature
Geometry

• What are the
undefined terms in
geometry?
• What concepts
present the
foundations of
geometry?
• Can you sketch the
intersection of
lines and planes?
QUESTIONS!

THE UNDEFINED
TERMS IN
GEOMETRY

The terms points, lines, and planes are the
foundations of geometry, but point, line, and
plane are all what we call undefined terms.
How can that be?
Any definition we could give them would depend
on the definition of some other mathematical idea
that these three terms help define. In other words,
the definition would be circular!
Undefined Terms?

The Three Undefined Terms

• Has no dimension
• Indicates location
• Usually represented by a small dot
A
A point is represented with a capital letter.
The above is called point A.
Point

• Extends in one dimension.
• Represented with a straight line with two
arrowheads to indicate that the line extends
without end in two directions.
Line

Naming a Line
l
A
B
Line
1. Using a small letter
(e.g. l or m)
2. Using two points on
the line
The given line can be named as
AB (line AB) l (line l)
Note that the letters can be interchanged, so we have BA

• Extends in two dimensions.
• Represented by a slanted 4 sided figure,
but you must imagine it extends without
end, even though the representation has
edges.
Plane

A plane can be named using a capital letter
or three points on the plane
A
B C
M
This is plane M or plane ABC
Plane

Undefined Terms: A Summary
Undefined
Term
Represen-
tation
Naming
Other
Information
Point Dot
•
• P
Read: point P
Has no length, width
or thickness;
occupies no space
Line Straight mark
with two arrow
heads
Two capital letters
with a double
arrowhead above
them or a lower
case letter
Has infinite length
but no width and
thickness; is
straight
Plane Slanted four-
sided figure
Single capital letter
Read: Plane M
Has infinite
length and width
but no thickness;
is a flat surface
M

Determine the undefined term
suggested by each of the following.
1. the tip of a pencil
2. the top of the teacher’s table
3. telephone wires
4. the wall of your classroom
5. the surface of the page of your book
Check your
understanding
point
plane
line
plane
plane

Determine the undefined term
suggested by each of the following.
6. the ruler’s edge
7. the tip of a needle
8. a corner of a piece of paper
9. the floor of your bedroom
10. the string on a guitar
Check your
understanding
line
point
point
plane
line

ADDITIONAL CONCEPTS
RELATED TO THE
UNDEFINED TERMS

• Collinear points are points that lie on the
same line.
l
C
A
B Points A, B and C are collinear.
Collinear and Coplanar Points
• Coplanar points are points that lie on the
same plane.
A
B
C Points A, B and C are coplanar.

Examples
COLLINEAR POINTS
- B, A, L, K
- S, M, K
- Q, M, ?
- T, ? , D S
M
Q
B
D
H
T
K
LA
D or B
A or S

NONCOLLINEAR
POINTS
- B, A, M, K
- D, M, K
- Q, M, ?
- T, ? , D
S
M
Q
B
D
H
T
K
LA
Examples
Any point except B and D
Any point except A and S

Examples
COPLANAR POINTS
- G, E, O
- F, L, I
- O, E, ?
- A, ? , C
G or B
F, L, or I

Examples
NONCOPLANAR
POINTS
- F, A, G
- G, O, A, L
- O, E, ?
- A, ? , C
Any point except G or B
Any point except F, L, or I

1. 2 points that are collinear.
2. 3 points that are coplanar.
3. 2 points that are noncollinear.
M
G
D F
B
Check your
understandingGiven the figure shown, name the following:
4. 3 points that are noncoplanar.

A B D
What happens if
you cut the line
shown below at
points B and C?
What figures will
you form?
C
Subsets of a Line

A B
DCA B
C DB C
RAY RAYSEGMENT
Subsets of a Line

B
A LINE SEGMENT (or segment) is a set of points
consisting of two points on a line, and all the
points on the line between the two points
C
LINE SEGMENT
Subsets of a Line
The two fixed points at the end of a line
segment are called ENDPOINTS.

A line segment is named by its endpoints.
The points can be interchanged in the name
B C
This is line segment BC or CB
In symbols:
BC or CB
Subsets of a Line

A RAY is the part of the line which consists
of a fixed point called the ENDPOINT and
the set of all points on one side of the
endpoint.
The endpoint is the starting point and the
arrow determine the direction of the ray.
A B
RAY
Subsets of a Line

A
ray BA but not ray AB
DB C
In naming a ray, the endpoint always
comes first, followed by the second
point.
ray CD but not ray DC
BA CD
Subsets of a Line

Not all symbols are created equal!
Symbol alert!
BAAB is the same as
AB BAis the same as
BUT…
A B
A B
Subsets of a Line
The ray is different!

AB is NOT the same as BA
A B AB
A B BA
Initial point
1st
Notice that the initial point is listed first in the symbol.
Also note that the symbolic ray always has the arrowhead
on the right regardless of the direction of the ray.
Symbol alert!
Subsets of a Line

then and are opposite rays.
If C is between A and B,
Opposite Rays
CA
A BC
CB
C is the common initial point for the rays.
Subsets of a Line

• Two or more geometric figures intersect if they
have one or more points in common.
• The intersection of the figures is the set of
points the figure has in common
How do 2 lines intersect?
How do 2 planes intersect?
What about a line and a
plane?
Intersection of Lines and Planes

To think about the questions on the last slide
let’s look at the following…
B A
ETwo lines
intersect at a
point, like here
at point A. F
Line BF is the intersection of the
planes G and H.
G
H
Point E is
the
intersection
of plane H
and line EC
D
C

If two lines intersect, then they intersect in
exactly one (one and only one) point.
The point exists (there is at least one point) and is
unique (no more than one point exists).
A

If two planes intersect, then their
intersection is a unique line.

A plane and a line not on the plane
intersect at a unique point.
E

Summing it up!

The Basics of Geometry 2:
Introduction to Postulates
and Theorems
What’s Next?

Math 7 geometry 01 undefined terms rev 2

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