I remember that once (Maybe $2000$ years ago), I asked this question on Physics SE, and the title of question was:
Can Newton's laws of motion be proved (mathematically or analytically) or are they just axioms?
There are many things I learnt from answers there abut the most thing was:
Asking whether one can prove something is a meaningless question unless one specifies the axioms one is allowed to use in the proof.
Whenever we see something in mathematical context we have two choices:
$1$. Assume it to be an axiom/postulate.
$2$. Prove it.
The first thing is easy to do because you have to do nothing, but it has also got some disadvantages. I mean you cannot be assured after choosing something as axiom because it can be wrong too, that is why we select only those things as axioms/postulates which are obvious.
Now, when we try to prove something, we do it by taking help of previously proved things or established axioms/postulates.
Now, let's come to your question, you are asking for a proof of equality alternate angles and whatever..
Now, it is quite right to think that way because yes you can prove that things provided that what axioms/postulates you want to assume true before constructing a proof.
For example: You can prove that alternate angles are equal by assuming that angles between two parallel lines are supplementary and a straight lines is just $180$ degrees (In terms of angles ).
I shall let you conclude.