Math 7 geometry 02 postulates and theorems on points, lines, and planes

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Basics of Geometry 2Postulates and Theorems Related

to Points, Lines and Planes

Foundations of Geometry

Definitions

Undefined Terms

Postulates and Theorems on Points, Lines, and Planes

Postulates

TheoremsEuclid

Father of Geometry

DefinitionsDEFINITIONS are words that can be

defined by category and characteristics

that are clear, concise, and reversible.


Example: Definition of Line Segment

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

PostulatesPOSTULATES statements accepted

as true without proof.

They are accepted on faith alone.

They are considered self-evident

statements.


They are also called AXIOMS.

Ruler Postulate

PART 1: There is a one-to-one

correspondence between the points of a

line and the set of real numbers.

This means that every point on the number

line corresponds to a UNIQUE real number


Ruler Postulate

PART 2: the distance between any two points equals the absolute value of the difference of their coordinates.

a b

a bDistance =


Segment Addition Postulate

If B is a point between A

and C, then

AB + BC = AC

A B C

Note that B must be on AC.


Definition of “Betweenness”

If A, B, and C are points

such that AB + BC = AC,

then B is between A and

C.

A B C



Examples


1. has length 10 cm and has

length 8 cm. If A is between P and

K, find the length of

PA AK

PK


Examples


2. Is on a number line and O is

between B and X. If the

coordinates of B and O are 3 and

8, respectively, and BX = 12, what

is the coordinate of X?

BX

The Midpoint of a Line Segment


The MIDPOINT of a line segment

is a point that divides the segment

into two equal segments.

A M B

M is the midpoint of AB

1

2AM MB AB

Examples


3. has length 10 cm. If J is the

midpoint of , what are the

lengths of the following?

KL

KL

a. KJ b. JL


Examples


4. Find the coordinate of the

midpoint of on the number line

if the coordinates of L and N are –3

and 7, respectively.

LN


Line Postulate

Through any two points

there is exactly one line.

Restated: 2 points determine a unique line.


Plane Postulate

Part 1: Through any three

points there is at least one

plane.

Part 2: Through any three non-

collinear points there is exactly

one plane.


Three collinear points

can lie on multiple

planes.

M

While three non-

collinear points can lie

on exactly one plane.

(Three noncollinear

points determine a

unique plane)


Plane Postulate

With 3 non-collinear points, there is only one

plane – the plane of the triangle.B

A C


Plane Postulate

Flat Plane Postulate

If two points of a line are in a

plane, then the line

containing those points in

that plane. M

AB


Intersection of Planes Postulate

If two planes intersect, then their

intersection is a line.

Remember, intersection means points in common or in both sets.





H

GF

E

D

CB

A



Final Thoughts on Postulates

Postulates are accepted as true on

faith alone. They are not proved.

Postulates need not be memorized.

Those obvious simple self-evident

statements are postulates.

It is only important to recognize

postulates and apply them

occasionally.


Theorems

Theorems are important

statements that are proved

true.

We are not yet ready to learn


These are statements that

needs to be proven using

logical valid steps.

The principles and ideas used in proving theorems

will be discussed in Grade 8

Intersection of Lines Theorem

If two lines intersect, then they

intersect in exactly one point.

This is very obvious.

To be more than one the line

would have to curve.

But in geometry,

all lines are straight.


Theorem

Through a line and a point not on the line

there is exactly one plane that contains

them.

A


Restatement: A line and a point not on the line

determine a unique plane.

TheoremThrough a line and a point not on the line

there is exactly one plane that contains

them. WHY?

A

B C


If you take any two points

on the line plus the point off

the line, then…

The 3 non-collinear points

mean there exists a exactly

plane that contain them.

If two points of a line are in the plane, then line is in

the plane as well.

If two lines intersect, there is exactly one

plane that contains them.

Theorem


Restatement: Two intersecting lines determine a

unique plane.

If you add an

additional point from

each line, the 3

points are

noncollinear.

Through any three noncollinear points there is

exactly one plane that contains them.

If two lines intersect, there is exactly one

plane that contains them. WHY?

Theorem


Foundations of Geometry:

1 Undefined terms: Point, Line & Plane

2 Definitions

3 Postulates

4 Theorems

Statements accepted without proof.

Statements that can be proven true.

Primitive terms that defy definition due to circular definitions.

Words that can be defined by category and characteristics

that are clear, concise, and reversible.

Summing it up!Postulates and Theorems on Points, Lines, and Planes

Thank you!Thank

you!

Math 7 geometry 02 postulates and theorems on points, lines, and planes

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Transcript of Math 7 geometry 02 postulates and theorems on points, lines, and planes