Proposition 4

If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.

Let TeX Embedding failed!, TeX Embedding failed! be two triangles having the two sides TeX Embedding failed!,TeX Embedding failed! equal to the two sides TeX Embedding failed!,TeX Embedding failed! respectively, namely TeX Embedding failed! to TeX Embedding failed! and TeX Embedding failed! to TeX Embedding failed!, and the angle TeX Embedding failed! equal to the angle TeX Embedding failed!.

I say that the base TeX Embedding failed! is also equal to the base TeX Embedding failed!, the triangle TeX Embedding failed! will be equal to thr triangle TeX Embedding failed!, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that is, the angle TeX Embedding failed! to the angle TeX Embedding failed!, and the angle TeX Embedding failed! to the angle TeX Embedding failed!.

For, if the triangle TeX Embedding failed! be applied to the triangle TeX Embedding failed!,

and if the point TeX Embedding failed! be placed on the point TeX Embedding failed!

and the straight line TeX Embedding failed! on TeX Embedding failed!,

then the point TeX Embedding failed! will also coincide with TeX Embedding failed! because TeX Embedding failed! is equal to TeX Embedding failed!.

Again, TeX Embedding failed! coinciding with TeX Embedding failed!,

the straight line TeX Embedding failed! will also coincide with TeX Embedding failed!, because the angle TeX Embedding failed! is equal to the angle TeX Embedding failed!;

hence the point TeX Embedding failed! will also coincide with the point TeX Embedding failed!, because TeX Embedding failed! is again equal to TeX Embedding failed!

But TeX Embedding failed! also coincided with TeX Embedding failed!

hence the base TeX Embedding failed! will coincide with the base TeX Embedding failed!.

[For if, when TeX Embedding failed! coincides with TeX Embedding failed! and TeX Embedding failed! with TeX Embedding failed!, the base TeX Embedding failed! does not coincide with the base TeX Embedding failed!, two straight lines will enclose a space : which is impossible.

Therefore the base TeX Embedding failed! will coincide with TeX Embedding failed! and will be equal to it] [C.N. 4]

Thus the whole triangle TeX Embedding failed! will coincide with the whole triangle TeX Embedding failed!,

and will be equal to it.

And the remaining angles will also coincide with the remaining angles and will be equal to them,

the angle TeX Embedding failed! to the angle TeX Embedding failed!,

and the angle TeX Embedding failed! to the angle TeX Embedding failed!.

Therefore etc.

(Being) what it was required to prove.