My gf told me that his professor gave her homework. She says that the homework is not graded but will be asked in exam.
It starts of with an axiom.
Given a line and a point outside the line, there is one and only one line parallel to the original line that go through the point.
Try to proof:
- Transitivity of parallel lines
- Show that if line g and h is parallel and line k cross line g then line k will cross line h.
I think and think about it but I cannot solve it.
I am pretty sure there must have been other assumptions or axioms behind that. After all, that axiom is true in three dimensions but the theorems don't work on 3 dimensions.
I look up internet. I found that this is about Euclid plane geometry.
The playfair version of Euclid's 5th postulate is exactly the axiom in the problem.
If I can assume that lines are parallel if and only if they do not meet in plane geometry then I can derive those 2 theorems easily without that Playfair axiom. The exception is a trivial case if the line is the same.
I tried to "prove" that from Playfair's axiom and quora basically says that you can't prove that. Those are axioms in Euclid.
I check that Playfair's axiom is a bit like Euclid's 5th postulate and the other 4 isn't talking about axioms at all. Euclid talks about drawing circles and stuff instead of defining parallel lines.
Is there something I am missing?
Can we proof transitivity of parallel lines from Playfair's axiom?
Note:
There is a proof in transitivity property of parallel lines proof
That proof says that if two lines are not parallel then they must meet. To me, that sounds like another assumption or axiom. Can we use that? That's the problem.
That's the essence of my problem. I can't proof that without extra assumptions and I don't know what reasonable assumptions I can use.
Can I assume that lines do not meet (in plane) if and only if they're parallel?