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!
Ac1.2aHowToWriteDefinitionsGeometry1
Jun 27, 2015
Writing Definitions and Undefined Terms This is for high school students.
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GeometryThe 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!
Millau BridgeSir Norman Foster
Point, Lines, Planes, Angles
FallingwatersFrank Lloyd Wright
Millenium ParkFrank Lloyd Wright
Components of Geometry Part 1
Components of Mathematical System
1. Undefined terms.
2. Definitions
Words and Statements
3. Postulates
4. Theorems
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
Undefined Terms
DescriptionA 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?
Point
Lines
DescriptionA 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
DescriptionA 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 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.
A
The letter in the corner is the name of the plane: plane A.
Why Undefined terms?
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.
Circular Definitions
Geometric ExampleDefinition #1A point is the intersection of two lines.
Definition #2A 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.
RevisitedDefinition #1A point is the intersection of two lines.
Definition #2A line is a continuous series of points going in opposite directions.
Definition #1A point is the intersection of two lines.
Definition #2A line is a continuous series of points going in opposite directions.
But you say that you can see it.
What if you were blind?
DefinitionsThere 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 boatlong 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 boatpropelled 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 piecethat one wears on your wrist.
3 Clear: easily understood, non-complex.
4 concise: short as possible.
5 Reversible: if read backward then the definition still works.
A watch is… a time piecethat one wears on your wrist.
Yes
No!
Yes
3 Clear: easily understood, non-complex.
4 concise: short as possible.
5 Reversible: if read backward then the definition still works.
A watch is… a time piecethat one wears on your wrist.
Yes
Now, yes!
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 1We will practice step 1 - Categorize
A woman is …
A car is …
A sandal is …
A lake is …
A pencil is …
a female
a vehicle
a shoe
a body of water
a writing instrument
Let’s Practice Step 1We will practice step 1 - Categorize
A girl is …
A bus is …
A vase is …
A pot is …
A strait is …
a female
a vehicle
a container
a container
a poker hand
Let’s Practice Step 2We will practice step 2 - Characterize
A woman is …
A car is …
A sandal is …
A lake is …
A pencil is …
a female
a vehicle
a shoe
a body of water
a writing instrument
older than 17
motorized
with 3 or 4 wheels for 6 or less people.
open
large surrounded by land.
that leaves a carbon mark.
with top open to the air.
Let’s Practice Step 2We will practice step 2 - Characterize
A girl is …
A bus is …
A vase is …
A pot is …
A strait is …
a female
a vehicle
a container
a container
a poker hand
less than 17.
with 3 of a kind and 2 of a kind.
with a lid used for cooking.
for flower arrangements.
motorized that transports 12 or more people.
open
C’est fini.
Good day and good luck.