Explain AAS Congruence, Using Rigid Motions

This is pretty much the same as ASA Congruence. The only difference is that in AAS is that the side is not enclosed by the angles.

Consider $\triangle ABC$ and $\triangle DEF$. For the two to be congruent, they must be similar with a scale factor of $1$. If we know that two corresponding angles are equal, then by AA similarity, $\triangle ABC\sim\triangle DEF$. From similarity,

$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}$$

But, if two corresponding lengths are equal, say $AB=DE$,

$$1=\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}$$

which means all corresponding sides are equal! Therefore, the triangles are congruent.

AAS Congruence states that in two triangles if two pairs of corresponding angles are equal and one pair of corresponding sides is equal, then the two triangles are congruent.

Remark: If $\angle A=\angle D$, $\angle B=\angle E$, and $CA=FD$, $\triangle ABC\cong\triangle DEF$. Why does this fail if $\angle A=\angle D$, $\angle B=\angle E$, and $AB=EF$?

Answer: because the sides are not corresponding!