Geometry is the study of points, lines, planes, and spatial relationships. It uses undefined terms like points, lines, and planes as building blocks to build up definitions and prove theorems about spatial relationships. Postulates are accepted statements of fact that serve as the basis for logical deductions in geometry. Through postulates and theorems, geometry presents complex spatial concepts in an organized way by defining relationships between fundamental terms.

1. GEOMETRY
LANGUAGE OF

2. o When you hear “Geometry”,When you hear “Geometry”,
what comes to your mind first?what comes to your mind first?
o What makes the subject uniqueWhat makes the subject unique
compared to previous years’compared to previous years’
Algebra?Algebra?
o What are important skills neededWhat are important skills needed
to study Geometry effectively?to study Geometry effectively?
HOOKHOOK

3. PEERACTIVITY
Even Column Students: Client
Odd Column Students: Architect
Give the Architect a list a specification for
your preferred condominium floor layout
consisting of 2 bedrooms, 1 toilet and bath,
kitchen and living room.
Design your client’s preferred layout.

4. The need to visualize and model
objects, concepts or behavior results is
the need to derive and study patterns.
Why is there a need to study patterns?Why is there a need to study patterns?
What is the implication of studyingWhat is the implication of studying
patterns?patterns?

5. o How do you describe andHow do you describe and
visualize complex objects?visualize complex objects?
o Why do we draw figures ofWhy do we draw figures of
an object?an object?
??

6. PATTERNS andPATTERNS and
INDUCTIVE REASONINGINDUCTIVE REASONING

7. 22,, 44,, 77,, 1111 . . .. . .
What is the next three terms?What is the next three terms?
What is the rule in finding the next terms?What is the rule in finding the next terms?
ProblemProblem
AA

8. Inductive reasoning is reasoning that is
based on the patterns you observe. This form
of reasoning tells what the next terms in the
sequence will be. A conclusion reached using
inductive reasoning is called a conjecture. A
counterexample to a conjecture is an
example which will show that a conjecture is
incorrect.

9. ProblemProblem
AA

10. Derive the chemical and structural formula for
butane and hexane.
ProblemProblem
AA

11. Determine whether each conjecture isDetermine whether each conjecture is truetrue oror false.false.
Give a counterexample for any false conjecture.Give a counterexample for any false conjecture.
Given:Given: A is an integerA is an integer
Conjecture: Additive inverse of A is negativeConjecture: Additive inverse of A is negative
Given:Given: M is an AA StudentM is an AA Student
Conjecture: M lives in Antipolo CityConjecture: M lives in Antipolo City
Given:Given: X is a winged organismX is a winged organism
Conjecture: X is a birdConjecture: X is a bird
ProblemProblem
AA

12. Composition – Letters, Words, Phrase, Sentence,
Theme
Biology – Cells, Tissue, Organ, System, Organism
Society – Barangay, Town, Province, State, Union
Matter – Atoms, Molecules, Element, Compound,
Mixture
Can Geometry have the sameCan Geometry have the same
organizational structure?organizational structure?
What are the building blocks ofWhat are the building blocks of
Geometry?Geometry?

14. UNDEFINED TERMSUNDEFINED TERMS
Points, Lines and PlanesPoints, Lines and Planes

16. POINTPOINT

17. Point
• First undefined term
• No size and no dimension
• Merely a position
• A dot named with a capital letter
A

18. LINELINE

19. Line
• Second undefined term
• Consist of infinite number of points
extending without end in both directions
• Usually named with any two of its points or a
lower case letter
A B
k
AB
k

20. PLANEPLANE

21. Plane
• Third undefined term
• Represent a flat surface with no thickness
that extends without end in all directions
• Usually named by a capital letter or by three
points that are not on the same line

22. Plane
Q
E
W
R
plane Q or plane EWR

23. Based on the picture, Geometry is telling me
something. Share it.

24. DEFINED TERMSDEFINED TERMS
Space
Collinear
Coplanar
Intersection
Half Planes

25. SPACE is the set of all points.

26. Space
• Space is the set of all points

27. Space
• At least four noncoplanar points distinguish
space.
A
B
D
C

28. Collinear points are points that lie on
the same line.

29. Collinear Points
• Points are collinear if and only if they lie on
the same line.
– Points are collinear if they lie on the same line
– Points lie on the same line are collinear.

30. Collinear Points
A BC
D
• C, A and B are collinear.

31. Collinear Points
A BC
D
• Points that are not collinear are
noncollinear.

32. Collinear Points
A BC
D
• D, A and B are noncollinear.

33. Coplanar points are points that lie on
the same plane.

34. Coplanar Points
• Points are coplanar if and only if they lie on
the same plane.
– Points are coplanar if they lie on the same plane.
– Points lie on the same plane are coplanar.

35. Coplanar Points
• E, U, W and R are coplanar
• T, U, W and R are noncoplanar
E
W
R
T
U

36. Intersection is the set of points
common to two or more figures.

37. Intersection
• A set of points is the intersection of two
figures if and only if the points lie in
both figures

38. Intersection
A
B
C
k
Line k intersects CB at A

39. Half-planes
• Line n is contained in plane Q. Line n separates Q
into three sets of infinitely many points. One of the
sets is n itself. Two other are called half-planes . n is
the edge of each half-planes but is not contained in
either half plane.
S
T
Q
n
R

40. Half-planes
• S and R are on the same side of n and thus lie
on the same half-plane. S and T are on
opposite sides of n and thus lie in the
opposite half-planes.
S
T
Q
n
R

41. A postulate or axiom is an
accepted statement of fact.
Can we doubt a postulate?
Do we need to show validity of a
postulate?
What are some postulates in your beliefs?

42. GLENCOE TextGLENCOE Text

43. 1.1. Through any two points there is exactlyThrough any two points there is exactly
one line.one line.
2.2. If two lines intersect, then they intersectIf two lines intersect, then they intersect
in exactly on point.in exactly on point.
3.3. If two planes intersect, then theyIf two planes intersect, then they
intersect in exactly one line.intersect in exactly one line.
4.4. Through any three noncollinear pointsThrough any three noncollinear points
there is exactly one plane.there is exactly one plane.
Postulates from Prentice Hall
(Textbook)

44. Postulate 1
• A line contains at least two distinct points. A
plane contains at least three noncollinear
points. Space contains at least four
noncoplanar points.
ADDISONADDISON
WESLEYWESLEY

45. Postulate 2
P1 . A line contains at least two distinct points
P2. If two distinct points are given, then a
unique line contains them.
ADDISONADDISON
WESLEYWESLEY

46. Postulate 3
• Through any two points there are infinitely
many planes. Through any three points there
is at least one plane. Through any three
noncollinear points there is exactly one
plane.
ADDISONADDISON
WESLEYWESLEY

47. Postulate 4
• If two points are in a plane, then the line that
contains those points lies entirely in the
plane.
ADDISON-WESLEY TextADDISON-WESLEY Text
ADDISONADDISON
WESLEYWESLEY

48. Postulate 5
• If two distinct planes intersect, then their
intersection is a line.
ADDISON-WESLEY TextADDISON-WESLEY Text
ADDISONADDISON
WESLEYWESLEY

49. Theorem
• Using postulates as starting points, it is
possible to conclude that certain statements
are TRUE.
• Unlike postulates, theorems are statements
that must be proven true by citing undefined
terms, definitions, postulates, previously
proven theorems.

50. Theorems
• If there is a line and a point not in the line,
then there is exactly one plane that contains
them.
• If two distinct lines intersect, then they lie in
exactly one plane.

51. Existence and Uniqueness
• There exists at least one plane that
contains the intersecting lines.
(existence)
• There is only one plane that
contains the intersecting lines.
(uniqueness)

52. QUESTIONS
1. How many points are there in a line?
2. How many planes contain a single line?
3. How many planes pass through a single
point?
4. How many planes will contain three
noncollinear points?
5. How many planes will contain three
collinear points?

53. QUESTIONS
6. How many planes will contain two
intersecting lines?
7. How many planes will contain three
intersecting lines?
8. How points are there in a plane?
9. How many points of intersection between
two planes?
10.How many points do you need to define a
space?

54. Do AS 1, parts I, II, IV.Do AS 1, parts I, II, IV.
Textbook, p.14, nos. 48 – 51, HWJ SheetTextbook, p.14, nos. 48 – 51, HWJ Sheet
HOMEWork

55. In conveying ideas, what is theIn conveying ideas, what is the
advantage of presenting complexadvantage of presenting complex
concepts in organized fashion with well-concepts in organized fashion with well-
defined relationships?defined relationships?
JournalJournal