It’s wrong. We have systems that satisfy the axioms of non-Euclidean geometry, and so we know that the parallel postulate does not follow from the other axioms. Since you’re obviously interested in geometry, you should probably spend some time reading about them. There’s a really nice textbook that covers various geometries at an undergrad level, and I recommend it highly.

I would say you have smuggled in assumptions (about the truth of parallel postulate), which is easy to do given the subject matter, and that you should try and understand where in the proof such assumptions are made. And there is material in this thread which you can use to do so.

We have relative consistency proofs which show that Euclidean geometry is consistent iff hyperbolic geometry is consistent. If the parallel postulate follows from the first 4 axioms, then hyperbolic geometry is inconsistent, and thus Euclidean geometry is inconsistent. As un-intuitive as it might seem, and contra your project, proving that the parallel postulate follows from the first 4 axioms would prove that Euclidean geometry is inconsistent.

As **ultrafilter** says, it’s wrong. A good way to see why it’s wrong, as already recommended above, is to try to perform the construction in hyperbolic space, e.g. on the Poincare disk, and see where things go wrong. This is better done as an exercise than just having somebody show it to you, so I won’t draw the picture for you.