My geometry teacher told me this was supposed to be impossible

> assume the two lines extended meet opposite the side of which the interior angles have a sum less than two right angles
> because a set of interior and exterior angles are supplementary the sum of the interior and exterior angles must be the same as the sum of two right angles
> by the transitive property it is clear that on the side which the sum of the interior angles are less than two right angles the sum of the exterior angles must be greater than the sum of two right angles
> in each of the intersections the exterior angle of one side must be equal the interior angle of the other side
> by the transitive propertys it is clear that the sum of the interior angles on the side opposite the side of which the interior angles have a sum less than two right angles must be greater than the sum of two right angles
> because the three lines are distinct they must form a triangle when they intersect
> however this cannot happen due to the fact that the sum of the angles of a triangle is equal to the sum of two right angles which has already been exceeded by the interior angles of assumed to be intersecting side
> therefore euclid btfo

>have plane geometry problem
>frog post instead of using a diagram

the axiom in the book didnt have a picture so i assumed it wasnt necessary

They can intersect on the other side

> in each of the intersections the exterior angle of one side must be equal the interior angle of the other side
I have no idea what the fuck you're talking about

> because a set of interior and exterior angles are supplementary the sum of the interior and exterior angles must be the same as the sum of two right angles

> by the transitive property it is clear that on the side which the sum of the interior angles are less than two right angles the sum of the exterior angles must be greater than the sum of two right angles

- it is clear that on the side which the sum of the interior angles are less than two right angles
all of the sides have a sum interior angle of less than two right angles

I did not bother to read your thing in detail but the first few lines look like Euclid's original description of the fifth axiom.

Perhaps the paradox happens because the first few lines creates a setup that contradicts Euclid's fifth axiom and then goes on to find a triangle with angle sum bigger than 180.

I guess this would be a neat paradox to trick kiddies given that nowadays, no one teache Euclid's original axiom. Everyone just teaches its equivalent that the sum of the angles of every triangle must equal 180. This was in fact a theorem of Euclid, not an axiom. But well, whatever. Maybe I'm wrong as I did not care to carefully read your thing.

i forgot what its called when angles in an intersection are opposite of eachother

i asked my teacher if postulates and axioms from the textbook can be proved by contradiction

he said that it was a problem with the way that the textbooks are written and then showed me a copy of the elements and said that his five axioms are the basis for geometry and to basic to proof

i havent actually read the elements so i dont know the definitions and theorems he used

Ah well, if you care about resolving the issue you have then reading Euclid's fifth axiom would a good start. Compare this axiom to the statements given in the first lines of your post and see if you can find a contradiction. Or perhaps ask your teacher.