Chapter 1: Tools of Geometry

Chapter 1: Tools of Geometry

1- 3 Points, Lines, and Planes

Objectives, student will be able:

1. To understand basic terms of geometry. 2. To understand basic postulates of

geometry.

CW/HW: pgs. 19-21 #’s 2 – 48 Even

Points

How to sketch:

How to label:* Use capital printed letters.

* dotsA

B

CA

* Never label two points with the same name (in the same sketch). *Think of points as location markers. *Points are zero dimensional.

Collinear Points

Collinear points are points that lie on the same line. (The line does nothave to be visible.)

A

BC

non-collinearcollinear

A B C

Definition:

Lines

How to sketch:

How to label:

1- small “script” letter

* must have arrows at both ends

2- any two points on the line

(Two methods)

A B C

n

AB

BA

AC

CA

BC

CB

ABC

Possible Names:

??

??

?

?

Planes

How to sketch:

How to label:

* horizontal* vertical

1- capital “script” letter2- any three points on the plane

(Two methods)

* other

horizontal “edges”

vertical “edges”

M

PN

Think of a plane as a flat surface that has no thicknessA plane contains an infinite many lines and extends without end

Planes

How to label: #2- any three points on the plane

Planes

How to label: any three points on the plane

H

E

G

DC

BA

F

How many planes do you see?

Planes

How to label: any three (or more) points on the plane

H

E

G

DC

BA

F

How many planes do you see?

DC

BAplane ABCDplane ABCplane BCDplane CDAplane DABetc.

Planes

How to label: any three (or more) points on the plane

H

E

G

DC

BA

F

How many planes do you see?

plane EFGHplane EFGplane FGHplane GHEplane HEFetc.

H

E

G

F

Planes

How to label: any three (or more) points on the plane

H

E

G

DC

BA

F

How many planes do you see?

plane AEHDplane EHDplane HDAplane DAEplane AEHetc.

H

E

D

A

Planes

How to label: any three (or more) points on the plane

H

E

G

DC

BA

F

How many planes do you see?

plane BFGCplane BFGplane FGCplane GCBplane CBFetc.

G

C

B

F

Planes

How to label: any three (or more) points on the plane

H

E

G

DC

BA

F

How many planes do you see?

plane ABFEplane ABFplane BFEplane FEAplane EABetc.

E

BA

F

Planes

How to label: any three (or more) points on the plane

H

E

G

DC

BA

F

How many planes do you see?

plane DCGHplane DCGplane CGHplane GHDplane HDCetc.

H G

DC

Planes

How to label: any three (or more) points on the plane

H

E

G

DC

BA

F

Any three points determine a plane!

plane AGFplane BDGetc.

plane AFGD

G

D

A

F

plane ACGE

E

G

C

A

plane ACH

H

C

A

Coplanar

Coplanar objects (points or lines) are objects that lie on the same plane. (The plane does not have to be visible.)

Definition:

H

E

G

DC

BA

F

Are they coplanar?

ABC ? yesABCF ? NOHGFE ? yesEHCB ? yesAGF ? yesCBFH ? NO

Postulate or Axiom

A postulate or axiom is an accepted statement as fact.

Postulate and Axioms have no formal proof they exist or are true

Many mathematicians do years of research trying to prove postulates or axioms true.

Postulates and axioms that are proven true are known as Theorems.

Postulate 1-1

Through any two points there is exactly one line.

Line t is the only line that passes through points A and B.

Postulate 1-2

If two lines intersect they intersect in exactly one point

and intersect at C. AE BD

Postulate 1-3

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

Plane RST and Plane STW Intersect in .

AE

AE

ST

Postulate 1-4

Through any three noncollinear points there is exactly one plane.

“Think of a tripod”

Space

Space – the set of all points Total space in zero dimensions is a point. Total space in one dimension is a line. Total space in two dimensions is a plane Total space in three dimensions infinite in

all diections