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LANGUAGE OF
GEOMETRY
o When you hear “Geometry”, what
comes to your mind first?
o What makes the subject unique
compared to Algebra?
o What are important skills needed
to study Geometry effectively?
The need to visualize and model
objects, concepts or behavior results is
the need to derive and study patterns.
Why is there a need to study shapes and
patterns?
o How do you describe and
visualize complex objects?
o Why do we draw figures of
an object?
UNDEFINED TERMS
Points, Lines and Planes
•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!
Point
•
•
•
•
First undefined term
No size and no dimension
Merely a position
A dot named with a capital letter
A
Line
• Second undefined term
• Consist of infinite number of points
extending without end in both directions
• Usually named with any two of its points or a
lower case letter
k
AB
k
A
B
Plane
• Third undefined term
• Represent a flat surface with no thickness
that extends without end in all directions
• Usually named by a capital letter or by three
points that are not on the same line
Plane
Q
W
R
E
plane Q or plane EWR
DEFINED TERMS
Space
Collinear
Coplanar
Intersection
Half Planes
SPACE is the set of all points.
Space
• Space is the set of all points
Space
• At least four noncoplanar points distinguish
space.
B
D
A
C
Collinear points are points that lie on
the same line.
Collinear Points
• Points are collinear if and only if they lie on
the same line.
– Points are collinear if they lie on the same line
– Points on the same line are collinear.
Collinear Points
D
C
A
• C, A and B are collinear.
B
Collinear Points
D
C
A
• Points that are not collinear are
noncollinear.
B
Collinear Points
D
C
A
• D, A and B are noncollinear.
B
Coplanar points are points that lie on
the same plane.
Coplanar Points
• Points are coplanar if and only if they lie on
the same plane.
– Points are coplanar if they lie on the same plane.
– Points lie on the same plane are coplanar.
Coplanar Points
• E, U, W and R are coplanar
• T, U, W and R are noncoplanar
T
U
W
R
E
Line Segment
•Let’s look at the idea of a point in between two other points on a line.
Here is line AB, or recall
symbolically AB
This is segment
A
B
The line segment does not
extend without end. It has
endpoints, in this case A and B.
The segment contains all the
points on the line between A
and B
AB
Notice the difference in the
symbolic notation!
Ray
Let’s look at a ray:
A is called the initial
point
A
B
Ray AB extends in
one direction without
end.
•Symbolized by
AB
The initial point is
always the first letter
in naming a ray.
Notice the difference
in symbols from both
a line and segment.
Symbol alert!
•Not all symbols are created equal!
AB
is the same as
BA
A
B
AB
is the same as
BA
A
B
BUT…
Symbol alert!!
The ray is different!
AB
is not the same as
Initial point 1st
BA
A
B
A
B
AB
BA
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.
Opposite Rays
•If C is between A and B,
A
C
then CA and
B
CB are opposite rays.
C is the common initial point for the rays!
Angles
•Rays are important because they help us define something very important
in geometry…Angles!
•An angle consists of two different rays that have the same initial point.
The rays are sides of the angles. The initial point is called the vertex.
vertex
B
sides
A
C
Notation: We denote an angle with
three points and  symbol. The middle
point is always the vertex. We can also
name the angle with just the vertex
point. This angle can be denoted as:
BAC , CAB, or A
Classifying Angles
•Angles are classified as acute, right, obtuse, and straight,
according to their measures. Angles have measures greater than
0° and less or equal to 180°.
A
A
Acute angle
0°< m A < 90°
Right angle
m A = 90°
A
Obtuse angle
90°< m A < 180°
A
Straight angle
m A = 180°
Intersection is the set of points
common to two or more figures.
Intersections of lines and planes
• 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
Think!!
How do 2 line intersect?
How do 2 planes intersect?
What about a line and a plane?
Intersection
k
C
A
B
Line k intersects CB at A
Modeling Intersections
• To think about the questions on the last slide lets look at the following…
E
Two lines
intersect at a
point, like here at
point A.
A
B
F
D
H
C
G
Line BF is the intersection of the
planes G and H.
Point E is the
intersection
of plane H
and line EC
Intersection
• A set of points is the intersection of two
figures if and only if the points lie in
both figures
Based on the picture, Geometry is telling me
something. Share it.
QUESTIONS
1. How many points are there in a line?
2. How many planes contain a single line?
3. How many planes pass through a single
point?
4. How many planes will contain three
noncollinear points?
5. How many planes will contain three
collinear points?
QUESTIONS
6. How many planes will contain two
intersecting lines?
7. How many planes will contain three
intersecting lines?
8. How points are there in a plane?
9. How many points of intersection between
two planes?
10.How many points do you need to define a
space?
In conveying ideas, what is the
advantage of presenting complex
concepts in organized fashion with welldefined relationships?