Lesson No. 1

Alexander Marzonia Morron JrMathematics Educator

BNHS - GawaranEuclidean Geometry

Fundamental Ideas in

Geometry

Geometry

The word "geometry" comes from two Greek wordsgeo and metron Meaning "earth measuring."

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, …)

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?, ...)

Geometric Pictures

Millau BridgeSir Norman Foster

Point, Lines, and Planes

FallingwatersFrank Lloyd Wright

Millenium ParkFrank Lloyd Wright

Components of Geometry

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

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.

POINT

POINTA 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.

- It shows location. - It is named by one capital letter. A

B C

A, B, and C are POINTS

The pictures shown above represent a line.

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).

ILLUSTRATION

A B CThe opposite arrow indicates that it extends indefinitely in opposite direction.

LINES CAN BE NAMED IN 2 WAYS:1. BY TWO POINTS. HENCE, TWO POINTS DETERMINE A LINE.FOR EXAMPLE, IN THE FIGURE,

A B CLine AB or AB. Or Line BA or BA

Line AC or AC. Or Line CA or CA

2. LINES ARE ALSO NAMED WITH LOWERCASE LETTERS OR A SINGLE LOWER CASE LETTER.A B CThe line above can be

named also as line m instead of line AB or line AC.

m

COLLINEAR POINTS & NON-COLLINEAR POINTS

A B C

A, B, and C are COLLINEAR POINTS

COLLINEAR POINTS & NON-COLLINEAR POINTS

P

R

Q

P, Q, and R are NON-COLLINEAR POINTS

COLLINEAR POINTS-Points that lie on the same line.

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

Examples

COLLINEAR POINTS

- B, A, L, K- S, M, K- Q, M, ? - T, ? , D

S

M

Q

B

D H

T

KLA

Examples

NONCOLLINEAR POINTS

- B, A, M, K- D, M, K- Q, M, ? - T, ? , D

S

M

Q

B

D H

T

KLA

THE FLOOR

THE WALL AND THE CEILING

PICTURE FRAME

PLANEA 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.

β

-A plane has no thickness and edges.

- has length and width.

PLANE

In naming,-Named by 3 distinct points which are not collinear.

-For example, plane BDC.

D

B

C

PLANE

β

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

PLANE

HOW MUCH DO YOU KNOW…

• What are the basic terms of geometry?

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

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•

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• 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).

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• 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.

3 Undefined TermsA 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.

UNDEFINED TERMS

• Undefined terms are terms that are accepted without definition.

• The undefined terms we will use are: set, point, line and plane.

Geometry Leeson: Undefined Terms, Lines, Line Segments

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Aim: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.

What are the undefined terms?

Geometry Leeson: Undefined Terms, Lines, Line Segments

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Point:A point indicates a position or location in space.

. P

X

Y

. A(2, 6)

Points are named using capital letters and/or coordinates.

Geometry Leeson: Undefined Terms, Lines, Line Segments

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Line:A line is an infinite set of adjacent points.

Ex: Curved line

Ex: Straight line

Naming a Line:

a) Two points on the line:

b) Single lowercase letter

, , , , ...AB AC BA BC etc�������������������������������������������������������������������������������������������� �������

A B C

m

Geometry Leeson: Undefined Terms, Lines, Line Segments

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Plane: A plane is a set of points that forms a completely flat surface.

Naming a Plane:

a) Three points on the plane:

b) Single uppercase letter:A • C•

B •Plane ABC

R

Plane R

Geometry Leeson: Undefined Terms, Lines, Line Segments

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Def: Collinear Points: A collinear set of points is a set of points all of which lie on the same straight line.

A B C D

E

•Points A, B, C and D are collinear.

•Points A, E and C are not collinear.

Geometry Leeson: Undefined Terms, Lines, Line Segments

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Def: Line Segment:

Naming a Line Segment:

Use the names of the endpoints.A B

“Line segment” AB is part of “Line” AB�������������� �

A line segment is the set of two points on a line called endpoints, and all points on the line between the endpoints.

A B