Introduction to Geometry
1-2 Points, Lines, and Planes
Here’s what we’re going to do today
1. What is geometry?
2. 1-2 lesson
3. Assignment
What is Geometry?
is a Greek word
• “Geo” = “earth”
•
“-metry” = “process of measuring”
is the mathematical study of
Spatial Relationships
How geometry works
We start with
• a few simple building blocks
(points, lines, and planes),
• a number of simple statements that
we assume to be true
(
, or axioms),
, and
of numbers and figures
Then we use
• logical reasoning
to show that those starting points
inevitably lead to other true statements
(
and other proofs).
·
logical reasoning
and other proofs
Some Tools of Geometry
: Statement of the meaning of a word or phrase by
specifying necessary and sufficient conditions.
(we’ll talk about what this means later)
: Simple statements that are obviously true.
(these are our starting points for proving things)
: An opinion or conclusion based on incomplete
information. (unproven and may be wrong)
: A conjecture that has been proven true.
Undefined
of Geometry
(names, notation, and diagrams)
: a location; infinitesimally small (no size at all).
·
: is straight, has no thickness, extends forever in both
directions.
: is a flat surface, has no thickness, extends forever in
every direction along the surface.
The Basic Geometric Figures
Points, Lines, and Planes
These are abstract ideas we use to “model” real things.
Some examples are
• a point to represent a location of a city on a map
• a point to represent a corner (specific location) of a box
• a line to represent a corner edge where two walls meet
• a line to represent a state border
• a plane to represent the surface of a building face
• a plane to represent the surface orientation of the Earth’s
orbit around the Sun
Some
AC, denoted 𝐴𝐶, consists of points 𝐴 and 𝐶, and
all points that are between 𝐴 and 𝐶.
AC, denoted 𝐴𝐶, consists of 𝐴𝐶 and all other points 𝑃
such that 𝐶 is between 𝐴 and 𝑃.
have the same endpoint, but point in
opposite directions (they form a line).
More
:
:
:
:
What is the Intersection of…
• a line and a line:
• a line and a plane:
• a plane and a plane:
(Postulate 1-1)
Through any 2 points
·
·
there is exactly (only) 1 line
(Postulate 1-2)
If 2 distinct lines intersect,
·
then they intersect in exactly (only) 1 point
(Postulate 1-3)
If 2 distinct planes intersect,
then they intersect in exactly (only) 1 line
Do the 1-2 assignment in MathXL