Side-Side-Side Congruence Theorem

Consider the triangles $△ABC$ and $△DEF,$ where If there exists a rigid motion that maps one of these onto the other, then they are congruent. As the sides $\overline{AB}$ and $\overline{DE}$ are congruent, there is a rigid motion that maps one of these onto the other. Performing this transformation for one of the triangles leads to the two congruent sides overlapping. The line $\overleftrightarrow{C{F}^{\prime }}$ can now be drawn, dividing the angle $\angle C$ into $\angle 1$ and $\angle 3,$ and $\angle F^{\prime }$ into $\angle 2$ and $\angle 4.$ Notice that the $△BC{F}^{\prime }$ is an isosceles triangle, leading to $\angle 1$ and $\angle 2$ being congruent. Similarly, $\angle 3$ and $\angle 4$ are congruent as $△AC{F}^{\prime }$ is also an isosceles triangle. This leads to which by construction means that Thus, $\angle C$ and $\angle F^{\prime }$ are congruent. The triangles have two sides, and their included angle, that are congruent. Thus, by the SAS Congruence Theorem, the triangles are indeed congruent.