# proofs regarding intervals

• Mar 29th 2008, 07:28 PM
chocolateducttape
proofs regarding intervals
I feel bad asking for help because we're not really supposed to, but my professor isn't around on weekends and doesn't respond to emails, so if someone could just get me started that would be amazing.

What I have:
Axiom 1 - At least two points belong to each line.
Axiom 2 - Given any line, there is at least one point that does not belong to that line.
Axiom 3 - There exists at least one line.
Axiom 4 - Given two different points, there exists exactly one line that contains those two points.
Axiom 5 - If ABC, then A, B, and C are three different points of some line, and CBA.
Axiom 6 - If A, B, and C are three different points of a line, then one and only one of the following holds: ABC, BCA, CAB.
Axiom 7 - If A, B, C, and D are four different collinear points and ABC, then one and only one of the following is true: ABCD, ABDC, ADBC, DABC.
Axiom 8 - If A and B are two different points, then there is a point C such that ABC; there is a point D such that ADB; and there is a point E such that EAB.
Definition of an interval: Let D and H be any two different points. Then the interval DH is the set of points D, H, and all points between D and H.
Theorem 14: Each interval has at least three points.
Theorem 15: YZ = ZY.
The proof for Theorem 14:
Let AB be an interval. By the definition of an interval, A and B are two different points. By Axiom 8, if A and B are two different points, then there is a point C such that ACB. Therefore, each interval must contain at least three points. (And I never would have figured that out if I hadn't spent the entire class period on it yesterday and reworked it five times until the professor finally laughed and told me how to do it.)

What I don't have:
The proof for Theorem 15.
Do I start by saying "Let YZ and ZY be intervals"? Or "Let YZ be an interval"? Or "Let Y and Z be two different points"? Or something else entirely? Apparently we have to let something be something, but I don't know what that ought to be. Should I use the proof from Theorem 14 and then add Axiom 5 to that?

I miss high school geometry, all we did in that class was draw pictures and I had an A the whole year.
• Mar 30th 2008, 12:28 AM
Opalg
Quote:

Originally Posted by chocolateducttape
Theorem 15: YZ = ZY.
...
Do I start by saying "Let YZ and ZY be intervals"? Or "Let YZ be an interval"? Or "Let Y and Z be two different points"? Or something else entirely? Apparently we have to let something be something, but I don't know what that ought to be. Should I use the proof from Theorem 14 and then add Axiom 5 to that?

I think that the statement of Theorem 15 assumes that YZ is an interval. So I would start by saying "Since YZ is an interval, Y and Z are different points." Then (as you suggest) use Theorem 14 and Axiom 5 to show that if YXZ then ZXY and vice versa.