I'm going through Euclid's Elements (Book 1) (Proposition 4). We're given two triangles (ABC, DEF) where two sides of the first triangle are equal to the two other sides (say AB=DE and AC=DF) of the second triangle. The angle between these two sides are also equal so (A=D).
In the proof, we place vertex A onto vertex D. Because the angles are equal and because we have the sides equal, we can conclude that the vertices B and C coincide with the vertices E and F respectively. This is fine so far.
Now, given that the three vertices coincide with the three other vertices, can we just use axiom 8 (magnitudes which coincide with one another or exactly fill the same space are equal) to conclude that the triangles are equal and so the remaining sides and angles are also equal?
In Byrne's version, we first use axiom 10 (two straight lines cannot include a space) to conclude that the sides BC and EF are equal. I was wondering if we really need this axiom or not?