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.
Best Answer
How much you need to write to justify any statement in a proof depends a lot on context. If you are writing for an audience, you have to know how much they can fill in the details.
If you are writing homework for a course, your instructor should tell you how much detail to include.
I tell my students to write enough to convince me that they have convinced themselves for good reasons - convincing me is not enough since I already know what's true. In an advanced course it's OK to skip explanations for steps that are essentially elementary and "obvious". In an introductory course I expect to see complete arguments. If a student writes too much I can point out what's unnecessary.