1. Lesson BNHS -
No. 1 Gawaran
Euclidean
Geometry
Fundamental Ideas in
Geometry
Alexander Marzonia Morron Jr
Mathematics Educator

2. Geometry
The word "geometry" comes
from two Greek words
geo and metron
Meaning "earth measuring."

3. Geometry is the visual study of shapes, sizes,
patterns, and positions.
It occurred in all cultures, through at least one
of these five strands of human activities:
1. building/structures
(building/repairing a house, laying out a garden, making a kite, …)
2. machines/motion
(using a pry-bar, riding a bike, sawing a board, swinging, …)
3. navigating/star-gazing
(How do I get from here to there?, using maps, …)

4. Geometry is the visual study of shapes, sizes,
patterns, and positions.
It occurred in all cultures, through at least one
of these five strands of human activities:
4. Art/patterns
(designs, symmetries, representations, …)
5. Measurement
(How big is it?, How far is it?, ...)

7. Millau Bridge
Millenium Park Sir Norman Foster
Fallingwaters
Frank Lloyd Wright Frank Lloyd Wright
Point, Lines, and Planes
Components of Geometry

8. Three undefined terms:
Points, Lines, and Planes. (P L P)
• We can give descriptions of these
three terms.
• We also use these terms to help us
write definitions of other terms such
as segment, or ray

9. WHY THESE ARE CALLED
UNDEFINED?
It is because they can only be
defined circularly.
Which means in terms of each
other or in terms of themselves.

11. POINT
A point is usually represented by a
dot on a piece of paper, a pinhole, and
the tip of a sharp needle.
It is so small that it can never be
measured because it has no dimensions
at all.

12. - It shows location.
- It is named by one capital letter.
A
B C
A, B, and C are POINTS

13. The pictures shown above represent a
line.

14. LINES
A line is always straight and travels forever
(INFINITE) in two directions.
A line may be represented by the edge of a ruler
or a piece of stretched thread.
has no width BUT IT HAS length( which is
infinite).

15. ILLUSTRATION
A B C
The opposite arrow indicates that
it extends indefinitely in opposite
direction.

16. LINES CAN BE NAMED IN 2 WAYS:
1. BY TWO POINTS.
HENCE, TWO POINTS DETERMINE A
LINE.
FOR EXAMPLE, IN THE FIGURE,
Line AB or AB. Or Line BA or
BA A B C
Line AC or AC. Or Line CA or
CA

17. 2. LINES ARE ALSO NAMED WITH
LOWERCASE LETTERS OR A SINGLE
LOWER CASE LETTER.
m
A B C
The line above can be
named also as line m instead
of line AB or line AC.

18. COLLINEAR POINTS & NON-
COLLINEAR POINTS
A B C
A, B, and C are COLLINEAR
POINTS

19. COLLINEAR POINTS & NON-
COLLINEAR POINTS
Q
P
R
P, Q, and R are NON-
COLLINEAR POINTS

20. COLLINEAR POINTS
-Points that lie on the same line.
NONCOLLINEAR POINTS
- Points that do not lie on the same
line.

21. Examples
COLLINEAR POINTS T
A L
- B, A, L, K B K
- S, M, K D H
- Q, M, ? M
- T, ? , D
S Q

22. Examples
NONCOLLINEAR POINTS T
A L
- B, A, M, K B K
- D, M, K D H
- Q, M, ? M
- T, ? , D
S Q

23. THE FLOOR

24. THE WALL AND THE CEILING

25. PICTURE FRAME

26. PLANE
A plane is often represented by a FLAT
SURFACE. These plane' surfaces are used to
connect any two or more points on a
straight line.
The walls of the classroom, the top of a
table, and the surface of the chalkboards
are examples of plane.

27. -A plane has no
thickness and
edges.
- has length
and width. β

28. In naming,
-Named by 3
distinct points
which are not
collinear.
D
-For example,
plane BDC. C
B

29. or
-Single greek
alphabet.
-For example,
Plane β
β

30. HOW MUCH DO YOU KNOW…
• What are the basic terms of geometry?

31. QUIZ
• Determine the undefined term suggested by each of the
following.
• 1. the tip of a pencil
• 2. the top of a coffee table
• 3. telephone wires
• 4. the wall of a room
• 5. the surface of the page of a book

32. QUIZ
• Determine the undefined term suggested by each of the
following.
• 6. the ruler’s edge
• 7. the tip of a needle
• 8. a window pane
• 9. the floor of your bedroom
• 10. the string on a guitar
•

33. • Euclidean Geometry - History:
• Euclid was the first mathematician of his time to use
the axiomatic method.
• Euclid’s axiomatic system consisted of four parts:
undefined terms, which lead to definitions, which
lead to postulates (also called axioms, which are
accepted as true), which lead to theorems (which are
proven by deductive reasoning).
33

34. • Euclidean Geometry – History:
• The undefined terms on which Euclid based his
system were point, line, and plane.
• Point – a location in space
• Line – a straight arrangement of points
• Plane – a two dimensional surface that extends
infinitely in both directions ( i.e., a table top )
• Euclid used the undefined terms to introduce
certain definitions as they were needed in his
axiomatic system.
34

35. 3 Undefined Terms
A point is like a star in the night sky. It is a little spec in a large endless sky. However, unlike
starts, geometric points have no size. Think of them as being so small that they take up zero
amount of space.
A line is like the edge of a ruler, that never ends. Take a ruler and draw a line - now imagine if
that line kept going straight forever. The line you have is thick enough for you to see, but you
need to imagine that your line is so thin that you can't see it - it has no thickness at all. That is
a geometric line.
A plane is a flat piece of land (like a football field) that extends forever. Imagine that you can
pick that football field up, and put it anywhere in the air that you like. You can even turn it
side ways, or diagonally. A plane is just a flat edge (like a piece of paper) that has no
thickness (just like the line) and extends forever.

36. UNDEFINED TERMS
• Undefined terms are terms that are accepted
without definition.
• The undefined terms we will use are:
set, point, line and plane.

37. Aim: What are the undefined terms?
Do Now:
1) What is a point?
2) What is a line?
3) What is a plane?
The most fundamental concepts in geometry do
not have precise definitions but, we understand
their meanings intuitively. We say these concepts
are undefined.
Geometry Leeson: Undefined Terms, Lines,
37
Line Segments

38. A point indicates a position or location
Point:
in space.
.
Y
P . A(2, 6)
X
Points are named using
capital letters and/or
coordinates.
Geometry Leeson: Undefined
38
Terms, Lines, Line Segments

39. A line is an infinite set of adjacent points.
Line:
Ex: Curved line
Ex: Straight line
Naming a Line:
   
  
a) Two points on the line: A B , A C , B A , B C , etc ...
A B C
b) Single lowercase letter
m
Geometry Leeson: Undefined Terms, Lines,
39
Line Segments

40. Plane: A plane is a set of points that forms a
completely flat surface.
Naming a Plane:
a) Three points on the plane: B •
Plane ABC C •
A •
b) Single uppercase letter:
Plane R
R
Geometry Leeson: Undefined Terms, Lines,
40
Line Segments

41. Def: Collinear Points: A collinear set of points is a set
of points all of which lie on the
same straight line.
E
A B C D
•Points A, B, C and D are collinear.
•Points A, E and C are not collinear.
Geometry Leeson: Undefined Terms, Lines,
41
Line Segments

42. Def: Line Segment: A B
A line segment is the set of two points on a
line called endpoints, and all points on the line
between the endpoints.
Naming a Line Segment:
Use the names of the endpoints.
A B


“Line segment” A B is part of “Line” AB
Geometry Leeson: Undefined Terms, Lines,
42
Line Segments