1.5 Postulates and Theorems Relating Points, Lines, and Planes

1.5 POSTULATES AND THEOREMS RELATING POINTS, LINES, AND

PLANES

Conduct Experiments 1-3 Write down the goal for each experiment

in your notes Answer every question; that is something

that has a question mark.

Experiment 1Goal

To find out how many points determine a line.

Materials

Pencil Paper Straight edge

Procedure1. Draw two points on a piece of paper. To draw a point make a dot on the paper as if you were dotting the letter i.

2. Draw a line through these two points. Now draw another line through these same two points.

Is that even possible? Draw a third line through these two points. Are the

three lines the same of different?

3. Draw another two points. How many different lines can you draw through these two points?

Something else to think about

How many lines can you draw through a single point ?

Experiment 2Goal To find out how many points determine a

plane.

Materials play-doh Four sharpened pencils with erasers Tabletop

Procedure1. Form a ball of clay (your point) the size of a

small orange.

2. Stick the pointed end of two sharpened pencils (your lines) one inch into the ball of clay.

3. Turn the ball of clay so that both erasers touch the table (your plane) at the same time. Is it stable?

4. Remove the pencils and place them point first into the ball of clay in a different way. Turn the ball so that both erasers touch the table at the same time. Is it stable?

5. Stick a third pencil into the ball of clay.

6. Turn the ball of clay so that all three erasers touch the table at the same time. Is it stable?

7. Remove all three pencils from the ball of clay and place them point first in the ball of clay in a different way. Find a way to touch all three erasers to the table at the same time. Is it stable?

8. Now stick four sharpened pencils in the ball of clay. Is it possible to turn the ball of clay so that all four erasers touch the table at the same time?

9. Place the four pencils in the ball of clay so that is it impossible for all four erasers to touch the table at the same time. Is it stable?

Something to think about How many legs must a stool have so that

it does not tip over?

How many legs must a stool have to guarantee it always makes solid contact with the floor?

Did you ever sit in a wobble chair? Why was that?

Experiment 3Goal

Discover the properties of lines in a plane

Materials

Two pencils A table

Procedure1. Place two pencils on a table.

The table represents a plane.

2. Let the two pencils represent two straight lines, but remember that straight lines extend infinitely. Make an X with the pencils. The pencils now intersect at exactly one point.

3. Try to make the pencils intersect at more than one point. It’s impossible! The only way to make two pencils intersect at more

than one point is to put them on top of each other, but then they represent the same line.

Something to think about

How many different ways can two lines intersect?

SPECIAL WORDING “EXACTLY ONE” Unique No more than one “One and only one”

“AT LEAST ONE” Can be more than one Not a unique answer

You will see this wording in the postulates and

theorems and it is important to understand

Postulates and Theorems

Postulates and Axioms Statements that are accepted without

proof.

Postulate A line contains at least two points; a

plane contains at least three points not all in one line; space contains at least four points not all in one plane

The is sort of a broad postulate, while the others become more specific

Experiment 1 = Postulate Through any two points there is

exactly one line

• What does this postulate mean?• You can draw many lines through a single point

• Two points determine a single line• You can draw one line through any two different points

Experiment 2 = Postulate Through any three points there is at least

one plane, and through any three non collinear points there is exactly one plane.

What does this postulate mean?• Through three points that lie in a straight line

there are an infinite number of planes. (with a point there is infinite lines, with a line, there are infinitive planes)

• Through three points that do not lie in a straight line there is only one plane. (exactly one)

Postulate If two points are in a plane, then the line

that contains those points is in that plane.

Why? Because through any two points there is

exactly one line

Postulate If two planes intersect, then their

intersection is a line.

Theorems Important statements that are proved. We use postulates to prove theorems.

Experiment 3 = Theorem If two lines intersect, then they intersect

in exactly one point.

**We know this to be true because there was no way to make the pencils intersect twice **

Theorem Through a line and a point not in the line

there is exactly one plane.

Think triangles….(demo)

Theorem If two lines intersect, then exactly one

plane contains the lines.

Just think…no matter how I twist or turn these lines, I can always lay them down on a flat surface (the plane)

Remote time

True or False A postulate is a statement assumed to be

true without proof.

True or False The phrase “exactly one” has the same

meaning as the phrase “one and only one.”

True or False Through any two points there is exactly

one plane.

True or False Through a line and a point not on the line

there is one and only one plane.