Geometry
The word "geometry "
comes from two Greek words
geo and metron meaning "earth measuring."
Geometry was extremely important to ancient societies and
was used for surveying, astronomy, navigation, and building.
Geometry, as we know it is actually known as Euclidean
geometry which was written well over 2000 years ago in
Ancient Greece by Euclid, Pythagoras, Thales, Plato and
Aristotle just to mention a few. The most fascinating and
accurate geometry text was written by Euclid, and was called
Elements. Euclid's text has been used for over 2000 years!
Millenium Park
Frank Lloyd Wright
Millau Bridge
Sir Norman Foster
Fallingwaters
Frank Lloyd Wright
Point, Lines, Planes, Angles
Components of Geometry Part 1
Components of
Mathematical System
Words and Statements
1. Undefined terms.
2. Definitions
3. Postulates
4. Theorems
Undefined Terms
In geometry, definitions are formed using
known words or terms to describe a new
word. There are three words in geometry
that are not formally defined. These three
undefined terms are point, line and plane.
PLP: Point, Line and Plane
Point
Description
A point has dimensions zero,
a line is one-dimensional,
a plane two-dimensional, and
space is three-dimensional.
What might be a four-dimensional concept?
Point
POINT (an undefined term)
In geometry, a point has no dimension
(actual size). Even though we
represent a point with a dot, the point
has no length, width, or thickness. A
point is usually named with a capital
letter. In the coordinate plane, a point
is named by an ordered pair, (x,y).
Do you get the point?
Lines
Description
A line is a series of consecutive points going in
opposite directions. It is of infinite length. Like a
point it has no thickness. It is represented as a
line with arrows on each end to indicate the line
keeps going.
Planes
Description
A plane is a flat surface that extends outward in all
directions. It can be thought of as a set of
consecutive lines extending in opposite directions.
Planes
A
A plane is usually indicated by a parallelogram.
Although edges are drawn, they are not really
there. The figure is only a model of the plane.
The plane extends outward infinitely.
The letter in the corner is the name of the plane: plane A.
Why Undefined terms?
Circular Definitions
How do you define the first word without other words?
Have you ever looked up a word that was explained in
terms of another word that you did not know?
Have you looked up the new word to find that it was
defined in terms or the first word?
This can be a real problem.
In foreign languages this happens all the time.
This also happens in geometry.
Geometric Example
Definition #1
A point is the intersection of two lines.
Definition #2
A line is a continuous series of points going
in opposite directions.
What is wrong?
1 You need to know what a line is to define a point.
2 You need to know what a point is to define a line.
This is a never ending cycle.
Revisited
Definition #1
A point is the intersection of two lines.
Definition #2
A line is a continuous series of points going
in opposite directions.
Definition #1
A point is the intersection of two lines.
Definition #2
A line is a continuous series of points going
in opposite directions.
But you say that you can see it.
What if you were blind?
Definitions
There are a lot of terms in geometry – 7 pages of them.
There are too many definitions to memorize.
Therefore, you need to understand
how to write definitions.
If you have this skill,
your understanding
will enable you to define any term
without having to memorize.
A description is not a definition.
A definition gives the
minimal characteristics
that include all possible
variations or elements.
To define terms there is a 5 step process:
1 Categorize
2 Characterize
3 Clear
4 concise
5 Reversible
We will start with non-geometric terms.
To define terms there is a 5 step process:
1 Categorize: put the term in
2 Characterize
3 Clear
4 concise
5 Reversible
We will start with non-geometric terms.
1 Categorize: put the term into a
category that restricts
what you are talking about.
A canoe is…
a boat
We will start with non-geometric terms.
2 Characterize: list the minimum
characteristics to
A canoe is… a boat
long thin open
propelled by paddles.
3 Clear: easily understood, non-complex.
4 concise: short as possible.
5 Reversible: if read backward
then the definition still works.
A canoe is… a long thin open boat
propelled by paddles.
Yes to all three.
3 It is easily understood. 4 No words can be left out.
5 When read backwards it is complete and works.
Again we will start with non-geometric terms.
1 Categorize: put the term into a
category that restricts
what you are talking about.
A watch is…
a time piece
2 Characterize: list the minimum
characteristics to
A watch is…
a time piece
that one wears
on your wrist.
3 Clear: easily understood, non-complex.
Yes
4 concise: short as possible.
Yes
5 Reversible: if read backward
then the definition still works.
No!
A watch is…
a time piece
that one wears
on your wrist.
3 Clear: easily understood, non-complex.
Yes
4 concise: short as possible.
Yes
5 Reversible: if read backward
then the definition still works.
A watch is…
a time piece
that one wears Now,
on your wrist. yes!
Or in your pocket.
Note: that when read backwards
it was noted that pocket watches were left out.
Reversibility is the test
for a complete definition.
Note: that all definitions start the same way.
A __________ is …
A house is …
A circle is …
A parallelogram is …
Let’s Practice Step 1
We will practice step 1 - Categorize
A woman is … a female
A car is …
A sandal is …
A lake is …
a vehicle
a shoe
a body of water
A pencil is … a writing instrument
Let’s Practice Step 1
We will practice step 1 - Categorize
A girl is …
A bus is …
a female
a vehicle
A vase is …
a container
A pot is …
a container
A strait is …
a poker hand
Let’s Practice Step 2
We will practice step 2 - Characterize
A woman is … a female
A car is …
older than 17
a vehicle with 3 or 4 wheels
for 6 or less people.
motorized
A sandal is …
a shoe with top open to the air.
open
A lake is …
a body of water
large
surrounded by land.
A pencil is … a writing instrument
that leaves a carbon mark.
Let’s Practice Step 2
We will practice step 2 - Characterize
A girl is …
a female
A bus is …
a vehicle that transports
motorized
12 or more people.
A vase is …
a container for flower
open
A pot is …
less than 17.
a container
arrangements.
with a lid used
for cooking.
A strait is … a poker hand with 3 of a kind
and 2 of a kind.
C’est fini.
Good day and good luck.