Proposition I-4.

Proposition I-4. If two triangles have the two sides equal to two sides respectively, and have the angles contained by the sides equal, trhey 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.
Proof.

Since B coincides with E and C coincides with F, it follows that BC equals EF.

[Otherwise the two straight lines would enclose a space, which is impossible.]

Let ABDand DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively.

Assume also that angle BAC equals angle EDF.

We will first show that BC is equal to FE.

Let the triangle ABC be applied to the triangle DEF with the point A placed on the point D and the straight line AB placed on the line DE.

Then the point B will coincide with the point E.

Because the angle BAC is equal to the angle EDF it follows that the line AC equals the line DF.

Therefore, the point C coincides with the point F.

It follows that the whole triangle ABC coincides with the whole triangle DEF.

Therefore, they are equal.

Finally, this implies that the respective angles are equal; that is angle ABC equals angle DEF and angle ACB equals angle DFE.

QED