5th postulate of Euclid Geometry

'If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.'
The parallel postulate is a theorem and can be proven?
discuss

There's a formal justification for this which has to do with the nature of angles, and uses either Al Kashi's theorem or something with cosines and sines.

But you can verify it empirically, by defining any axis, and showing that the projection of both lines on that axis is converging, and regularily so (because they are straight lines and not curves).

I suck at explaining this, but that's pretty much it.

>>1
No, the parallel postulate is NOT a theorem, in fact it is a postulate (= a thing you make up and take for granted because it seems obviously true but for some reason you cannot prove it.)

If the parallel postulate were a theroem, which means that it could be proven starting from the roots of Geometry (= the more basic postulates), then there could NOT exist any Geometry with the same roots but without the parallel theorem, since it would descends from it.

The fact that those Geometries exist (Riemann's, etc.) are an indirect proof (proof! not 'hint') that the parallel postulate is just that.

Did I answer your question?

>>3
yes, thank you  :]