Many geometry books used to teach high-schoolers these days try to transfer Hilbert's reworking of Euclid's axioms into a (somewhat) palatable form for students. They don't usually seem to go into as much detail, say, about betweenness, but they include an SAS "Postulate" rather than an SAS "Theorem."
In particular, this book by Jurgensen, Brown, and Jurgensen has the following eleven postulates in its first three chapters (I could have them a little off):
- Segment addition
A line contains two points; a plane contains three noncollinear points; space contains four noncoplanar points
- Two points define a line
- Given three points, there exists a plane containing them; three noncollinear points determine a plane
- If a plane contains two points, then it contains the line connecting them
If two planes intersect, their intersection is a line
If a transversal crosses parallel lines, then corresponding angles are equal
- If a transversal crosses two lines and corresponding angles are equal, then the lines are parallel
and goes on to state the following theorem without proof:
3-11. If two lines are parallel to a third line, then the lines are parallel.
In my tutoring sessions with geometry students, we always use this book because I feel that it's pretty good. But when we get to this point, we go, "Okay, we're pretty competent at geometry; let's try to prove it." And we always breeze through the case where the lines are in a plane together, and come up dry in the general case.
Looking into Euclid XI tonight, I see that this is the ninth proposition. Its proof uses the sixth:
"If lines make right angles to the same plane, then they are parallel."
This, in turn, uses I.4 (SAS) and I.8 (SSS), which in Jurgensen/Brown/Jurgensen are postulates in the next chapter.
So is 3.11 provable from the eleven postulates? Or are the authors just careless--is there a model of postulates 1 through 11 in which there are two noncoplanar lines parallel to a third line?
Some definitions for this context.
Two lines are parallel if they are coplanar and nonintersecting.
A transversal is a line in a plane intersecting two other lines in that plane at different points.
A line and a plane are perpendicular if the line is perpendicular to all lines in the plane through the point of intersection.
I'm also not sure if the following (from Chapter 4) can be proved without any triangle postulates: "If a line is perpendicular to two lines in a plane, then it is perpendicular to all lines in that plane through the point of intersection."