This interesting question touches on a serious question about the axioms for Euclid's geometry. The parallel lines axiom (often called the "parallel postulate") seems to have a flavor different from the others. For centuries mathematicians tried to prove it. Along the way they discovered many theorems that are equivalent to it - you could use any one of them as an axiom instead of the parallel postulate and end up with the same geometry. Among those theorems:
The angles of a triangle sum to two right angles.
There is a pair of similar triangles that are not congruent.
Two lines parallel to the same line are parallel to each other.
The Pythagorean theorem.
The wikipedia page linked above lists more. I recommend it.
Eventually Lobachevski and Gauss and Bolyai and others discovered that you could do nice geometry even when the parallel postulate failed - thus discovering (or inventing) non-Euclidean geometry.
Edit in answer to a comment.
Did non-Euclidean geometry prove or disprove the postulate?
The answer is "neither". What the invention of non-Euclidean geometry proved is that it is impossible to prove or disprove the parallel postulate starting from the other axioms. More formally: if it's possible to reach a contradiction from the other axioms along with the negation of the parallel postulate then that contradiction can be reached from the other axioms and the parallel postulate.
The Greeks found the parallel postulate pretty clearly "true" in the "real world", so they built it into their abstraction of that world -
the Euclidean plane. But it did bother mathematicians from then on, hence the attempts to prove it and the eventual proof that you can't.
We don't in fact know if the parallel postulate is true in the space we live in. Einstein's theory of general relativity says it's not when matter is present. But even where matter is relatively rare, space may be curved in a sense that mathematicians have made precise. If it is curved then you have to look at a pretty large volume to tell - much as you have to look at a large area of the surface of the earth to detect that you are not on a Euclidean plane. (The Greeks did know that.) Search for is our world Euclidean to read more.