Q:

Can I Construct Anything (Within Reason) When Building a Proof?

I apologize if this is a silly question, but I'm curious about this. Is it acceptable to claim anything (as long as it's logically sound) during construction when building a geometric proof? For example, let's say I have $\triangle$ ABC and $\triangle$ DEF. Let's say I had some givens and wanted to prove equivalence for these two triangles. What if I wanted to say there's some point G that when connected to B makes an angle congruent to the angle created by some point H connected to E? Is it logically acceptable to just state that the lines I created produced two angles that are equivalent to each other? Or do I have to prove that these angles are equivalent? I guess  I'm just unclear about where the line is drawn (no pun intended) for building ancillary statements when constructing a proof. Again, apologies if this is a silly question.
EDIT 1: I'm trying to prove that $\triangle$ ABC $\cong$ $\triangle$ DEF with the following givens:

$\angle$ A $\cong$ $\angle$ D
Segment AC $\cong$ Segment DF

I had the idea of saying that there was some point G connected to point B, and some point H connected to point E, connected in such a way that their angles were identical. I then wanted to declare/construct a point X that is an altitude and median for $\triangle$ GAB and a point Y that is also an altitude and a median for $\triangle$ DEH. I then went on to say that because $\triangle$ GAB and $\triangle$ DEH were isoceles, segment AB $\cong$ segment DE. That along with Givens 1 and 2 would have been my SAS proof.

A:

A statement that something exists is itself a mathematical statement. On a formal level, if in the middle of a proof you assert the statement that some object exists satisfying some properties, then like any other step of a proof you must justify that statement. Sometime your justification will be an axiom, sometimes it will be an earlier construction, or maybe another theorem which has already be proved to be true.
As for writing your own proofs, I would say that your first audience is yourself: Are you convinced that you know the correct justification?
After that, generally speaking nothing is accepted as "self-evident" in mathematics. However, one should always keep one's readers or listeners in mind. If your reader/listener is a teacher, you have to know what standards they expect: for instance, probably your teacher won't require you to cite the very simplest axioms every time they are used to justify something in a proof. If your reader/listener is a peer or classmate, you have to have a feel for what they understand: if you and they have a common pool of understanding about the simplest principles of proof, again probably you do not have to justify things which fall within that common pool.
But here's a few experiences you will have if you pursue mathematics far enough: You will think some step of a proof is obvious, but your listener won't, and will demand a justification, and so you better hope you're ready to satisfy them! Or, you will think some step is obvious, but later on down the line you yourself will begin to have deep suspicions about the course of your proof, you will trace those suspicions back to the "obvious" step, which all of a sudden will stop being obvious, and you will really, really want to know whether that step can be justified.

