The proposition 6 says:
If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.
And the proof of Euclid in his Elements basically is (for detailed proof visit the link): Let's assume the sides opposite are not equal, then this allows us to construct a congruent triangle with the initial triangle (by the SAS criterion) inside of the initial triangle, but is not overlapping and this is absurd because the whole is greater than the part (Common notion 5). So there must be a contradiction assuming that in a triangle the sides opposite of congruent angles are not equal, thus they must be equals as requiered.
But imagine this Triangle △ABC where the ∡A = ∡B = 0
Following Euclid construction we obtain the congruent (by SAS) inside triangle △ABD with △ABC :
But this time both triangles do overlap completely.
Maybe I'm not completely 'formal' with the use of overlap but what else is supposed to mean in the contradiction of Euclid?
This is the first proof by contradiction given in the Elements and I do not have any against this method in general, but in this case it seems is not enough because it relies in what you can see in the diagram, and this could be very tricky, often they (the diagrams) have implicit/hidden hypothesis that you work with but they are not mentioned in the original statement.
And personally if the case is that the contradiction is you can 'see' that in the two congruent triangles one is smaller than the other, why he previously gave such a complicated construction in an earlier proposition just for copy one segment in another position? Yes, I know because to avoid "implicit hypothesis" like motion in the figures, and that's perfect but why soften the methods later? also in previous Prop. 5, he gave again such a complicated construction (well known as the Pons Asinorum) that he perfectly could have simplified but he rather choose the "more formal" one.
This proof in my opinion is out of the spirit of his work. Or maybe I'm not understanding his reasoning correctly.