expalining definition, theorem, and postulate

1. Definition, Theorem, and
Postulate
Meeting 1

3. Undefined terms : Point, line,
and plane
It can’t be defined but described
Definition
Words that can be defined by
category and characteristics that
are clear, concise, and reversible.
Postulates
Statements accepted without
proof.
Theorems
Statements that can be proven
true.
GEOMETRY

4. THE SOURCE OF
THE TRUTH

5. UNDEFINED TERMS

6. UNDEFINED TERMS (CONT)

7.  Garis ℓ dan garis 𝐴𝐵 terletak di
bidang 𝑄.
 Garis 𝑘 dan garis 𝐴𝐵 terletak
di bidang 𝑃.
 Garis 𝐴𝐵 merupakan garis
yang terletak pada
perpotongan bidang 𝑃 dan 𝑄.
Garis 𝐴𝐵 disebut garis
persekutuan kedua bidang
tersebut.
UNDEFINED TERMS (CONT)

8. DEFINED TERMS

9. DEFINED TERMS (CONT)

10. NAMING ANGELS
1. Using three letters, the center letter corresponding to the vertex of
the angle and the other letters representing points on the sides of
the angle. For example, the name of the angle whose vertex is 𝑇 can
be angle 𝑅𝑇𝐵 (∡𝑅𝑇𝐵) or angle 𝐵𝑇𝑅 (∡𝐵𝑇𝑅).

11. NAMING ANGELS
2. Placing a number at the vertex and in the interior of the angle. The
angle may then be referred to by the number. For example, the
name of the angle whose vertex is 𝑇 can be ∡1 or ∡𝑅𝑇𝐵 or ∡𝐵𝑇𝑅.

12. NAMING ANGELS
3. Using a single letter that corresponds to the vertex, provided that
this does not cause any confusion.
There is no question which angle on the diagram corresponds to
angle A, but which angle on the diagram is angle D? Actually three
angles are formed at vertex D:
• Angle 𝐴𝐷𝐵
• Angle 𝐶𝐷𝐵
• Angle 𝐴𝐷𝐶

13. Line
• it's in a straight path.
• goes in both directions.
• does not end ... so you can't measure it's length.
Ray
• it's straight.
• is part of a line.
• has one endpoint.
• goes in ONE direction.
Line Segment
• is straight.
• is a part of a line.
• has 2 endpoints that show the points that end the line.

14. EXAMPLE 1
a. Name the accompanying line.
b. Name three different segments.
c. Name four different rays.
d. Name a pair of opposite rays.

15. EXAMPLE 2
Use three letters to name each of the numbered angles in the
accompanying diagram.

16. Do you think there is another definition in geometry?
Apakah dalam geometri hanya istilah-istilah tersebut
yang didefinisikan?

17. DEFINITIONS
The purpose of a definition is to make the meaning of a term clear. A
good definition must:
• Clearly identify the word (or expression) that is being defined.
• State the distinguishing characteristics of the term being defined,
using only words that are commonly understood or that have been
previously defined.
• Be expressed in a grammatically correct sentence.

18. DEFINITIONS OF COLLINEAR AND
NONCOLLINEAR POINTS
Points 𝐴, 𝐵, and 𝐶 are collinear.
Points 𝑅, 𝑆, and 𝑇 are not collinear.
DEFINITION:
• Collinear points are points that lie on the same line.
• Noncollinear points are points that do not lie on the
same line.

19. DEFINITION OF TRIANGLE
A triangle is a figure formed by connecting three noncollinear points
with three different line segments each of which has two of these
points as end points.

20. Contoh 3
Susun konsep-konsep berikut dalam urutan pendefinisian:
• Segitiga samakaki, segitiga, sudut alas segitiga samakaki
Segitiga, segitiga samakaki, sudut alas segitiga samakaki
• Sisi miring, segitiga, segitiga siku-siku
Segitiga, segitiga siku-siku, sisi miring

21. A good definition must be reversible as shown in the
following table.
The first two definitions are reversible since
the reverse of the definition is a true
statement.
The reverse of the third “definition” is false
since the points may be scattered.

22. Contoh 4
• Segitiga siku-siku adalah segitiga dengan satu sudutnya siku-siku.
(benar)
Konvers:
Segitiga dengan salah satu sudutnya siku-siku adalah segitiga
siku-siku. (benar)
DEFINISI
• Setiap sudut siku-siku adalah sudut-sudut kongruen (sama besar)
(benar)
Konvers:
Sudut-sudut yang kongruen (sama besar) adalah sudut siku-
siku. (salah)
BUKAN DEFINISI

23. Contoh 5
Susunlah konsep-konsep berikut dalam urutan pendefinisan
• Ukuran sudut, sudut, sudut kongruen.
• Kaki segitiga samakaki, segitiga samakaki, segitiga
• Sudut, segitiga tumpul, sudut tumpul.
----
• Sudut, ukuran sudut, sudut kongruen
• Segitiga, segitiga samakaki, kaki segitiga samakaki
• Sudut, sudut tumpul, segitiga tumpul

24. Contoh 6
Manakah yang merupakan definisi?
1. Segitiga samasisi adalah segitiga di mana ketiga sisinya sama
panjang.
Definisi
2. Pada segitiga siku-siku, sisi miring adalah sisi di hadapan sudut siku-
siku.
Definisi

25. Postulates
Are statements accepted as true
without proof.
They are accepted on faith alone.
They are considered self-evident statements.

26. Some Important Terms
• Exists-there is at least one
“chairs exist in this room”
• Unique-there is no more than one
“In this room, the computer is unique, the chairs are not”
• One and only one-exactly one; shows existence and uniqueness
“In this room, there is one and only one fire extinguisher”

27. INITIAL POSTULATES
In building a geometric system, not everything can be proved since there
must be some basic assumptions, called postulates (or axioms), that are
needed as a beginning.
Postulate 1.1 implies that through two points exactly one line may be drawn
while.

28. INITIAL POSTULATES
Postulate 1.2 asserts that a plane is defined when a third point
not on this line is given.

29. #1 Ruler Postulate
• A] The points on a line can be paired with the real
numbers in such a way that any two points can have
coordinates 0 and 1.
We know this as the number line.
0- 4 -2 642
Whole numbers and fractions are
not enough to fill up the points on a line.
The spaces that are missing are filled by the irrational numbers.
43
2 , 3, 7, 11, ,etc

30. #1 Ruler Postulate
• B] Once a coordinate system has
been chosen in this way, the
distance between any two points
equals the absolute value of the
difference of their coordinates.
This is the more important part.
a b
a bDistance =

31. # 2 Segment Addition Postulate
B is between A and C so
AB + BC = AC
A B C
Note that B must be on AC.

32. #3 Protractor Postulate
• On AB in a given plane, chose any point O
between A and B. Consider OA and OB
and all the rays that can be drawn from O
on one side of AB. These rays can be
paired with the real numbers from 0 to
180 in such a way that:
• OA is paired with 0. and OB is paired with
180.
• If OP is paired with x and OQ with y, then
m POQ x y  

33. Relax! You don’t have to memorize this.
Restated:
1] All angles are measured between 00 and 1800.
2] They can be measured with a protractor.
3] The measurement is the absolute values of the
numbers read on the protractor.
4] The values of 0 and 180 on the protractor
were arbitrarily selected.

34. Protractor Postulate Cont.
0180
Q
P
B O A
x
y
m POQ x y  

35. #4 Angle Addition Postulate
• If point B is in the interior of , thenAOC
m AOB m BOC m AOC    
O A
B
C
1
2
1 2m m m AOC    

36. #4 Angle Addition Postulate
• is a straight angle and B is any
point not on AC , so
AOC
O A
B
C
0
180m AOB m BOC   
These angles are called “linear pairs.”
12
0
1 2 180m m   

37. Postulate #5
•A line contains at least 2 points;
• a plane contains at least 3 non-
collinear points;
• Space contains at least 4 non-
coplanar points.

38. Postulate #5
•A line is determined by 2 points.
• A plane is determined by 3 non-
collinear points.
• Space is determined by 4 non-
coplanar points.

39. Postulate # 6
•Through any two points there is exactly
one line.
Restated: 2 points determine a unique line.

40. Postulate # 7
•Through any three points there is at
least one plane.
•And through any three non-collinear
points there is exactly one plane.

41. Three collinear points can
lie on multiple planes.
M
While three non-collinear
points can lie on exactly
one plane.

42. Three collinear points can lie in multiple planes –
horizontal and vertical.

43. Three collinear points can lie in multiple planes –
Slanted top left to bottom right and bottom left to
top right.

44. With 3 non-collinear points, there is only one
plane – the plane of the triangle.
B
A C

45. Postulate # 8
• Two points of a line are in a plane and
the line containing those points in that
plane.

46. Notice that the segment
starts out as vertical with
only 1 pt. in the granite
plane.
As the top endpoint
moves to the plane…
The points in between
move toward the plane.
When the two endpoints lie in the plane
the whole segment also lies in the plane.

47. Postulate # 9
• The two planes intersect
and their intersection is a
line.
H
G
F
E
D
CB
A
Remember,
intersection means points in common or in both sets.

48. Postulate # 9
•The two planes intersect
and their intersection is a
line.
H
G
F
E
D
CB
A
Remember, intersection means points in common or in both sets.

49. Final Thoughts
• Postulates are accepted as true on faith alone. They are not proved.
• Postulates need not be memorized.
• Those obvious simple self-evident statements are postulates.
• It is only important to recognize postulates and apply them
occasionally.

50. Theorem 1.1
 If two lines intersect, then they intersect in exactly
one(one and only one) point.
The point exists(there is at least one point) and is unique(no more
than one point exists).
A .

51. Theorem 1.1
If 2 lines intersect, then they
intersect in exactly one point.
This is very obvious.
To be more than one the line
would have to curve.
But in geometry,
all lines are straight.

52. Theorem 1.2 (Know the meaning not the number)
 Through a line and a point not in the line, there is
exactly one(one and only one) plane.
The plane exists(there is at least one plane) and is unique(no more than one plane exists).
A
This is not so obvious.

53. Theorem 1.2
Through a line and a point not on the line
there is exactly 1 plane that contains them.
If you take any two points
on the line plus the point
off the line, then…
The 3 non-collinear points
mean there exists a exactly
plane that contain them.
If two points of a line are
in the plane, then line
is in the plane as well.
A
B C

54. Theorem 1.3 (Know the meaning not the number)
 If two lines intersect, then exactly one (one and only one) plane
contains the lines.
The plane exists(there is at least one plane) and is unique(no more than one plane exists).

55. Theorem 1.3
If two lines intersect, there is exactly
1 plane that contains them.
This is not so obvious.

56. Theorem 1.3
If two lines intersect, there is exactly
1 plane that contains them.
If you add an
additional point
from each line,
the 3 points are
noncollinear.
Through any three noncollinear points there is
exactly one plane that contains them.

57. Quick Quiz
 Two points must be ___________
Collinear
 Three points may be __________
Collinear
 Three points must be __________
Coplanar
 Four points may be __________
Coplanar

58. Quick Quiz
 Three noncollinear points determine a ___
Plane
 A line and a point not on a line determine
a __________
Plane
 A line and a plane can 1)__________
2)_________ or 3)____________
Be Parallel, Intersect in exactly one point, or
the plane can contain the line
 Four noncoplanar points determine
__________
Space

59.  For Kepler, a devout Christian, mathematics was itself a religious
undertaking. He wrote in Harmonice Mundi (1619):-
 Geometry existed before the creation; is co-eternal with the mind of
God; is God himself ... Where there is matter there is geometry. ...
geometry provided God with a model for the Creation and was
implanted into man, together with God's own likeness - and not merely
conveyed to his mind through the eyes. ... It is absolutely necessary
that the work of such a Creator be of the greatest beauty...

60. LATIHAN
1. Perhatikan gambar berikut.
a. Ada 6 segmen yang berbeda, sebutkan!
b. Ada 12 sinar yang berbeda, sebutkan!
c. Sebutkan nama garis di atas menurut 6 cara!
2. Sebutkan semua sudut yang tersebar pada gambar di bawah ini!

61. LATIHAN
3. Gunakan gambar berikut untuk mengisi pertanyaan di bawah.
a. 𝑇𝑅 ≅ ⋯
b. … ≅ 𝑌𝐴
c. … titik tengah 𝐴𝐵
4. Sebutkan nama-nama segitiga yang terdapat pada gambar berikut (ada 16 segitiga).
Kemudian kelompokan segitiga-segitiga tersebut berdasarkan jenis sudutnya (∆lancip, ∆ siku-
siku, ∆ tumpul).
S
Y

62. LATIHAN
5. Susunlah urutan istilah berikut menurut pendefinisiannya.
a. Segitiga samakaki, segitiga, titik puncak segitiga samakaki
b. Sudut-sudut kongruen, garis bagi sudut, sudut
6. Diketahui bidang I dan bidang II keduanya memuat titik 𝐴, 𝐵 dan 𝐶. Buktikan bahwa 𝐴, 𝐵 dan
𝐶 kolinear.