math 7 geometry 01 undefined terms rev 2

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A Mathematics 7 Lecture

The Basics of Geometry

QUESTION!

What words come to mind when you hear

the word

GEOMETRY?

The Basics of Geometry for High School

Geometry

OH MY!

Geometry Geometry comes from the Greek words geo (earth) and metron (measure), so geometry literally means “earth measure”Geometry is the branch of mathematics that defines and relates the basic properties and measurements of shapes

Geometry The ancient Egyptians, Chinese,

Babylonians, Romans, and Greeks used geometry for surveying,

navigation, astronomy, and other practical occupations. The Greeks

sought to systematize the geometric facts they knew by establishing

logical reasons for them and relationships among them.

The systematizing of geometric facts and

principles culminated in the geometry text

Elements, written in approximately 325 B.C.

by Euclid. This most remarkable text has been

in use for over 2000 years.

This recognizes Euclid as the Father of Geometry.

Euclid

Father of Geometry

Geometry

Geometry in Art and Nature

Geometry

• What are the undefined terms in geometry?

• What concepts present the foundations of geometry?

• Can you sketch the intersection of lines and planes?

QUESTIONS!


THE UNDEFINED TERMS IN

GEOMETRY


The terms points, lines, and planes are the foundations of geometry, but point, line, and plane are all what we call undefined terms.

How can that be?

Any definition we could give them would depend on the definition of some other mathematical idea that these three terms help define. In other words, the definition would be circular!

Undefined Terms?


The Three Undefined Terms


POINT

LINE

PLANE

• Has no dimension• Indicates location• Usually represented by a small dot

A

A point is represented with a capital letter.

The above is called point A.


Point

• Extends in one dimension.• Represented with a straight line with two

arrowheads to indicate that the line extends without end in two directions.

Line


Naming a Line

l

A

B

Line


1. Using a small letter (e.g. l or m)

2. Using two points on the line

The given line can be named as

AB (line AB) l (line l)

Note that the letters can be interchanged, so we haveBA9::::::::::::: :

• Extends in two dimensions.• Represented by a slanted 4 sided

figure, but you must imagine it extends without end, even though the representation has edges.

Plane


A plane can be named using a capital letter or three points on the plane

A

B C

M

This is plane M or plane ABC

Plane


Undefined Terms: A SummaryUndefined

TermRepresen-

tation Naming Other Information

Point Dot

•• P

Read: point P

Has no length, width or thickness; occupies no space

Line Straight mark with two arrow heads

Two capital letters with a double arrowhead above them or a lower case letter

Has infinite length but no width and thickness; is straight

Plane Slanted four-sided figure

Single capital letter

Read: Plane M

Has infinite length and width but no thickness; is a flat surface

M


Determine the undefined term suggested by each of the following.1. the tip of a pencil2. the top of the teacher’s table3. telephone wires4. the wall of your classroom5. the surface of the page of your book

Check your understanding

point

plane

line

plane

plane

Determine the undefined term suggested by each of the following.6. the ruler’s edge7. the tip of a needle8. a corner of a piece of paper 9. the floor of your bedroom10. the string on a guitar

Check your understanding

line

point

point

plane

line

ADDITIONAL CONCEPTS RELATED TO THE

UNDEFINED TERMS


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

l

CA

B Points A, B and C are collinear.


Collinear and Coplanar Points

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

A

BC Points A, B and C are

coplanar.

Examples

COLLINEAR POINTS

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

S

M

Q

B

D H

T

KLA



D or B

A or S

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

S

M

Q

B

D H

T

KLA

Examples



Any point except B and D

Any point except A and S

Examples

COPLANAR POINTS

- G, E, O- F, L, I- O, E, ? - A, ? , C



G or B

F, L, or I

Examples NONCOPLANAR POINTS

- F, A, G- G, O, A, L- O, E, ? - A, ? , C



Any point except G or B

Any point except F, L, or I

1. 2 points that are collinear.

2. 3 points that are coplanar.

3. 2 points that are noncollinear.

MGD F

B

Check your understandingGiven the figure shown, name the following:

4. 3 points that are noncoplanar.

A B D

What happens if you cut the

line shown below at

points B and C? What

figures will you form?

C


Subsets of a Line

QUESTION!

A B

DCA B

C DB C

RAY RAYSEGMENT


Subsets of a Line

B

A LINE SEGMENT (or segment) is a set of points consisting of two points on a line, and all the points on the line between the two points

C

LINE SEGMENT


Subsets of a Line

The two fixed points at the end of a line segment are called ENDPOINTS.

A line segment is named by its endpoints. The points can be

interchanged in the nameB C

This is line segment BC or CB

In symbols:

BC or CB


Subsets of a Line

A RAY is the part of the line which consists of a fixed point called the ENDPOINT and the set of all points on one side of the endpoint.

The endpoint is the starting point and the arrow determine the direction of the ray.

A B

RAY


Subsets of a Line

A

ray BA but not ray AB

DB C

In naming a ray, the endpoint always comes first, followed by the second point.

ray CD but not ray DC

BA::::::::::::::

CD::::::::::::::


Subsets of a Line

Not all symbols are created equal!

Symbol alert!

BAAB is the same as

AB BAis the same as

BUT…

A B

A B


Subsets of a Line

The ray is different!

ABis NOT the same asBAA B AB

A B BA

Initial point 1st

Notice that the initial point is listed first in the symbol. Also note that the symbolic ray always has the arrowhead on the right regardless of the direction of the ray.

Symbol alert!


Subsets of a Line

then and are opposite rays.

If C is between A and B,

Opposite Rays

CA

A BC

CB

C is the common initial point for the rays.


Subsets of a Line

• Two or more geometric figures intersect if they have one or more points in common.

• The intersection of the figures is the set of points the figure has in common

Think!!

How do 2 lines intersect?

How do 2 planes intersect?

What about a line and a plane?


Intersection of Lines and Planes

To think about the questions on the last slide let’s look at the following…

B A

ETwo lines intersect at a

point, like here at point

A.F

Line BF is the intersection of the planes G and H.

GH

Point E is the

intersection of plane H and line

ECD

C



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

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



A plane and a line not on the plane intersect at a unique point.



E

Summing it up!The Basics of Geometry for High School

The Basics of Geometry 2: Introduction to

Postulates and Theorems

What’s Next?


Thank you!

Thank you!

math 7 geometry 01 undefined terms rev 2

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