In The Non-Euclidean Revolution by Richard Trudeau, he discusses the theorems in Euclidean Geometry. In particular, I was struck by the proof of Theorem 23 (copying an angle). He avoids Euclid's proof because it used Theorem 8/SSS (which Euclid had proved with superposition). What he does instead:
Let $AB$ be the given line, $C$ a point on the line, and $\angle DEF$ the angle. (Hypothesis)
Case 1: If $\angle DEF$ is right,...
Case 2: If $\angle DEF$ is acute,...
Case 3: If $\angle DEF$ is obtuse,...
In each case he gives a construction that copies the angle. Since the three cases exhaust all possibilities then one can copy an angle.
From a construction point-of-view, I feel like this proof is incomplete since it doesn't provide a way to check if an angle is right, acute, or obtuse. I can't think of a proper way to tell the type of an angle that doesn't require observation. What if such a check was impossible? Doesn't this lead to a paradox? The paradox being that a theorem proves you can copy an angle but in reality you can't.