1.5 Postulates and Theorems Relating Points, Lines, and Planes

Conduct Experiments 1-4 Answer every question, that is something that has a question mark.

Experiment 1 Goal To find out how many points determine a line.

Materials Pencil Paper Ruler

Procedure 1. 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 2 Goal To find out how many points determine a plane.

Materials Clay Four sharpened pencils with erasers Tabletop

Procedure 1. 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. 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. 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. 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. 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? Remove all four pencils and place them point first in the ball of clay in a different way. Is there a way to touch all four erasers to the table at the same time? 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.

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 3 Goal Discover the properties of lines in a plane

Materials Two pencils A table

Procedure 1. 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?

“Exactly one” Unique No more than one One and only one

“At least one” Can be more than one Not a unique answer

Theorem Scramble

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

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 a single point there are an infinite number of planes. Through two points you can also find an infinite number of planes. Through three points that lie in a straight line there are an infinite number of planes. Through three points that do not lie in a straight line there is only one plane.

Postulate If two points are in a plane, then the line that contains those points is in that plane.

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

Postulates and Axioms Statements that are accepted without proof.

Theorems Important statements that are proved.

Experiment 3 = Theorem If two lines intersect, then they intersect in exactly one point.

Theorem Through a line and a point not in the line there is exactly one plane.

Theorem If two lines intersect, then exactly one plane contains the lines.

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 Three points determine a plane.

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.