Understanding Euclid's proof of SAS congruence

Question

In this proof, http://aleph0.clarku.edu/~djoyce/elements/bookI/propI4.html

I can't see why C.N.4 imply $\angle ABC = \angle DEF, \angle BCA= \angle EFD$

"Thus the whole triangle ABC coincides with the whole triangle DEF and equals it."

Can someone please detail this step?

Thanks.

Answer 1

All that CN 4 says is that whenever two things are "on top of each other" and line up perfectly, then they must be equal.

At the point in the proof when CN4 starts being used, we have placed triangle ABC "on top of" triangle DEF, and we have already seen the points A and D coincide, the points B and E coincide, and the points C and F coincide. Similarly, we already know AB coincides with DE, and AC coincides with DF.

So now we have 3 pairs of points which coincide, and 2 pairs of line segments which coincide. So it must be that the third sides of the triangles (BC and EF) also coincide, and by CN 4, since sides BC and EF coincide, those sides must be equal.

Now that means that triangle ABC and DEF coincide completely - all the points and line segments are in the same positions, so CN 4 tells us that if the triangles completely coincide, they must be equal.