Proposition 4

The fourth proposition proves SAS congruence. This means that if two triangles, △ABC and △DEF have two corresponding sides and the angles enclosed by them equal, then all corresponding sides and all angles of the two triangles must be equal. The way Euclid proves this fact is actually quite simple. Euclid imagines "superposing" D onto A so that the line DE lines up with the line AB. Since the length of DE equals the length of AB, this means E will coincide with B. Also, since the ∠CAB equals ∠FDE, then the line DF will line up with the line AC. Sine the length of DF equals the length of AC, this means F will coincide with C. Following this line of logic, all the vertices of △DEF will line up with their corresponding vertices of △ABC. This method is called superposition, and the logic behind it is a bit hazy. It's not clear what it even means to superpose the triangles onto each other, and Euclid doesn't have any postulate stating that this is a valid thing to do and that the concurrences he makes are legit.