1.5 Postulates and Theorems Relating Points, Lines, and Planes

1. 1.5 Postulates and Theorems Relating Points, Lines, and Planes

2. 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.

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

4. Materials • Pencil • Paper • Straight edge

5. 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?

6. Something else to think about • How many lines can you draw through a single point ?

7. Experiment 2 Goal • To find out how many points determine a plane.

8. Materials • play-doh • Four sharpened pencils with erasers • Tabletop

9. Procedure • Form a ball of clay (your point) the size of a small orange. • Stick the pointed end of two sharpened pencils (your lines) one inch into the ball of clay. • 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?

10. Stick a third pencil into the ball of clay. • Turn the ball of clay so that all three erasers touch the table at the same time. Is it stable? • 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? • 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? • 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?

11. 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?

12. Experiment 3 Goal • Discover the properties of lines in a plane

13. Materials • Two pencils • A table

14. Procedure • Place two pencils on a table. • The table represents a plane. • 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. • 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.

15. Something to think about • How many different ways can two lines intersect?

16. 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

17. Postulates and Theorems

18. Postulates and Axioms • Statements that are accepted without proof.

19. 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

20. 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

21. 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)

22. 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

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

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

25. 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 **

26. Theorem • Through a line and a point not in the line there is exactly one plane. • Think triangles….(demo)

27. 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)

28. Remote time

29. True or False • A postulate is a statement assumed to be true without proof.

30. True or False • The phrase “exactly one” has the same meaning as the phrase “one and only one.”

31. True or False • Through any two points there is exactly one plane.

32. True or False • Through a line and a point not on the line there is one and only one plane.