Download Geometry chapter 1-2, 1-3

Geometry chapter 1-2, 1-3
8-25-15
Postulates and Axioms
Axioms and postulates are accepted as true by definition. In Geometry,
we use definitions which are axiomatic, meaning the definition is
accepted as a fundamental truth.
Postulates are obvious consequences of these definitions.
For example, the statement, “Murder is wrong” is an axiom. We all just
know that purposely killing someone without a reason is wrong. We
don’t really need proof or an explanation of this.
Section 1-2
• Points
• Lines
• Planes
You will learn the symbols for each
You will learn the definitions of each
Section 1-2 continued
• Segments
• Rays
• Distance
You will learn about the symbols for segments and rays.
You will learn about how to measure and define segments.
You will learn about postulates.
Points
• A point represents a specific location in space. Points have no shape
or size.
• We represent a point using a capital letter and a dot.
D
Lines
• The straight path between two points is a line. A line extends forever
in both directions. Lines have no thickness and contain infinitely
many points.
k
B
A
• The notation for the above line is 𝐴𝐵. Notice it has arrows on both
sides. Sometimes a lower case letter is used to name a line.
Planes
• A plane is a flat surface the extends forever along two axes (think of
the coordinate plane on graph paper). It has no thickness and
contains infinitely many points and lines.
• Planes can be named using a single capital letter, or they can be
names after any three points on the plane that are not on the same
line.
The symbol for line
segments shows the
endpoints of the
segment with a line with
no arrows on it.
Line segments or
segments are finite
portions of lines. They
are identified by their
endpoints and can be
measured.
If we are referring to
the measure of the
line segment, we leave
off the line above the
endpoints.
Rays…
• A ray is a portion of a line that starts at one endpoint and continues
into infinity. In the notation of a ray, the endpoint is first. 𝐴𝐵
Identify 3 lines and 3
line segments
Postulates
• Now that we have some definitions, we can examine postulates.
• We will look at three postulates right now. Their order is not
important, different text books may use different ordering.
Postulate 1-1
• Through any two points, there is exactly one line.
• Can you imagine making more than one unique line (as defined in this
class) that passes through the same two points?
Postulate 1-2
• If two unique lines intersect, then they intersect in exactly one point.
• Try to draw an example of this in your notes.
• Can you think of this postulate ever not being true when you use the
geometric definition of a line?
Postulate 1-3
• If two distinct planes intersect, then they intersect in exactly one line.
• Imagine your desktop and notebook are each planes. If you hold the
edge of your notebook against the surface of the desk, the two meet
at the notebook’s edge. In what ways does this do a good or poor job
of modeling the postulate?
Postulate 1-4
• Through any three noncollinear points there is exactly one plane.
• What does collinear mean? What does noncollinear mean?
• Can you use this postulate to deduce the meaning of the words?
Postulate 1-5
• Postulate 1 – Ruler Postulate
The points on a line can be paired with real numbers in such a way that any
two points can have coordinates 0 and 1.
This means that I can choose two points on a line, label them 0 and 1, like on
a ruler.
Once a coordinate system has been chosen in this way, the distance between
any two points equal the absolute value of the difference of their
coordinates.
This means that once we establish the distance from 0 to 1 and therefore 1
to 2, 2 to 3 and so on, we have created a ruler we can use to measure length
In other words: “rulers work with segments”
Try these…