1. 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!

2. Point, Lines, Planes, Angles
Millau Bridge
Sir Norman Foster
Millenium Park
Frank Lloyd Wright
Fallingwaters
Frank Lloyd Wright
Point, Lines, Planes, Angles
Components of Geometry Part 1

3. Components of Mathematical System
Words and Statements
1. Undefined terms.
2. Definitions
3. Postulates
4. Theorems

4. Undefined Terms PLP: Point, Line and Plane
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

5. 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?

6. Point Do you get the 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?

7. 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.

8. 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.

9. 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.

10. Circular Definitions 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.

11. 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.

12. This is a never ending cycle.
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.

13. Revisited Definition #1 A point is the intersection of two lines.
A line is a continuous series of points going
in opposite directions.

14. But you say that you can see it.
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?

15. Definitions If you have this skill, your understanding
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.

16. minimal characteristics that include all possible
A description is not a definition.
A definition gives the
minimal characteristics
that include all possible
variations or elements.

17. 1 Categorize 2 Characterize 3 Clear 4 concise 5 Reversible
To define terms there is a 5 step process:
1 Categorize
2 Characterize
3 Clear
4 concise
5 Reversible

18. 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

19. We will start with non-geometric terms.
1 Categorize: put the term into a
category that restricts
what you are talking about.
a boat
A canoe is…

20. We will start with non-geometric terms.
2 Characterize: list the minimum
characteristics to
A canoe is…
a boat
propelled by paddles.
long thin open

21. 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 long thin open boat
propelled by paddles.
A canoe is…
Yes to all three.
4 No words can be left out.
3 It is easily understood.
5 When read backwards it is complete and works.

22. 1 Categorize: put the term into a category that restricts
Again we will start with non-geometric terms.
1 Categorize: put the term into a
category that restricts
what you are talking about.
a time piece
A watch is…

23. that one wears on your wrist. 2 Characterize: list the minimum
characteristics to
a time piece
A watch is…
that one wears
on your wrist.

24. No! 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 time piece
that one wears
on your wrist.
A watch is…

25. that one wears Now, on your wrist. yes! Or in your pocket.
3 Clear: easily understood, non-complex.
Yes
4 concise: short as possible.
Yes
5 Reversible: if read backward
then the definition still works.
a time piece
that one wears
on your wrist.
A watch is…
Now,
yes!
Or in your pocket.

26. Reversibility is the test
Note: that when read backwards
it was noted that pocket watches were left out.
Reversibility is the test
for a complete definition.

27. A __________ is … A house is … A circle is … A parallelogram is …
Note: that all definitions start the same way.
A __________ is …
A house is …
A circle is …
A parallelogram is …

28. Let’s Practice Step 1 A woman is … a female A car is … a vehicle
We will practice step 1 - Categorize
A woman is …
a female
A car is …
a vehicle
A sandal is …
a shoe
A lake is …
a body of water
A pencil is …
a writing instrument

29. Let’s Practice Step 1 A girl is … a female A bus is … a vehicle
We will practice step 1 - Categorize
A girl is …
a female
A bus is …
a vehicle
A vase is …
a container
A pot is …
a container
a poker hand
A strait is …

30. Let’s Practice Step 2 A woman is … a female older than 17 A car is …
We will practice step 2 - Characterize
A woman is …
a female
older than 17
A car is …
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
surrounded by land.
large
a writing instrument
A pencil is …
that leaves a carbon mark.

31. Let’s Practice Step 2 A girl is … a female less than 17. A bus is …
We will practice step 2 - Characterize
A girl is …
a female
less than 17.
A bus is …
a vehicle
that transports
12 or more people.
motorized
A vase is …
a container
for flower
arrangements.
open
A pot is …
a container
with a lid used
for cooking.
A strait is …
a poker hand
with 3 of a kind
and 2 of a kind.

32. C’est fini.
Good day and good luck.