01 fundamentals of geometry

GEOMETRYLANGUAGE OF

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 unique What makes the subject unique compared to previous years’ compared to previous years’ Algebra?Algebra?

o What are important skills needed What are important skills needed to study Geometry effectively?to study Geometry effectively?

HOOKHOOK

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.

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 studying What is the implication of studying patterns?patterns?

o How do you describe and How do you describe and visualize complex objects?visualize complex objects?

o Why do we draw figures of Why do we draw figures of an object?an object?

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PATTERNS and PATTERNS and INDUCTIVE REASONINGINDUCTIVE REASONING

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?

ProblemProblemAA

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.

ProblemProblemAA

Derive the chemical and structural formula for butane and hexane.

ProblemProblemAA

Determine whether each conjecture is Determine whether each conjecture is true true or or false. false. Give a counterexample for any false conjecture.Give a counterexample for any false conjecture. Given: Given: A is an integerA is an integerConjecture: Additive inverse of A is negativeConjecture: Additive inverse of A is negative

Given: Given: M is an AA StudentM is an AA StudentConjecture: M lives in Antipolo CityConjecture: M lives in Antipolo City

Given: Given: X is a winged organismX is a winged organismConjecture: X is a birdConjecture: X is a bird

ProblemProblemAA

Composition – Letters, Words, Phrase, Sentence, ThemeBiology – Cells, Tissue, Organ, System, OrganismSociety – Barangay, Town, Province, State, UnionMatter – Atoms, Molecules, Element, Compound, Mixture

Can Geometry have the same Can Geometry have the same organizational structure?organizational structure?What are the building blocks of What are the building blocks of Geometry?Geometry?

UNDEFINED TERMSUNDEFINED TERMSPoints, Lines and PlanesPoints, Lines and Planes

POINTPOINT

Point

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

A

LINELINE

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

kAB

k

PLANEPLANE

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

Plane

Q

E

W

R

plane Q or plane EWR

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

DEFINED TERMSDEFINED TERMSSpace CollinearCoplanarIntersectionHalf Planes

SPACE is the set of all points.

Space

• Space is the set of all points

Space

• At least four noncoplanar points distinguish space.

A

B

D

C

Collinear points are points that lie on the same line.

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.

Collinear Points

A BC

D

• C, A and B are collinear.

Collinear Points

A BC

D

• Points that are not collinear are noncollinear.

Collinear Points

A BC

D

• D, A and B are noncollinear.

Coplanar points are points that lie on the same plane.

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.

Coplanar Points

• E, U, W and R are coplanar• T, U, W and R are noncoplanar

E

W

R

T

U

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

Intersection

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

Intersection

A

B

C

k

Line k intersects CB at A

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.

ST

Q

n

R

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.

ST

Q

n

R

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?

GLENCOE TextGLENCOE Text

1.1. Through any two points there is exactly Through any two points there is exactly one line.one line.

2.2. If two lines intersect, then they intersect If two lines intersect, then they intersect in exactly on point.in exactly on point.

3.3. If two planes intersect, then they If two planes intersect, then they intersect in exactly one line.intersect in exactly one line.

4.4. Through any three noncollinear points Through any three noncollinear points there is exactly one plane.there is exactly one plane.

Postulates from Prentice Hall(Textbook)

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

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

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

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

Postulate 5

• If two distinct planes intersect, then their intersection is a line.

ADDISON-WESLEY TextADDISON-WESLEY Text

ADDISONADDISON

WESLEYWESLEY

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.

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.

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)

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?

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?

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

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

JournalJournal