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Rule

# Angle-Side-Angle Congruence Theorem

If two angles and the included side in a triangle are congruent with corresponding parts in another triangle, then the triangles are congruent.

This can be proven using rigid motions.

Consider the triangles $\triangle ABC$ and $\triangle DEF,$ where $\angle A \cong \angle D,\, \overline{AB} \cong \overline{DE}, \text{ and }\ \angle B \cong \angle E.$ If either of these can be mapped onto the other using rigid motion, then they are congruent. As $\overline{AB}$ is congruent with $\overline{DE},$ there is a rigid motion that maps one of these onto the other. This can be performed for one of the triangles, which leads to the two congruent sides overlapping.

Transform

The triangle $\triangle ABF'$ can now be reflected in the line $\overleftrightarrow{AB}.$ If the image of $F'$ falls onto $C,$ the triangles will completely overlap. As the angles $\angle CAB$ and $\angle F'AB$ are congruent, the ray $\overrightarrow{AF'}$ will be mapped onto the ray $\overrightarrow{AC}.$ Similarly, $\overrightarrow{BF'}$ will be mapped onto $\overrightarrow{BC}.$

Thus, the intersection of $\overrightarrow{AF'}$ and $\overrightarrow{BF'},$ which is $F',$ will be mapped onto the intersection of $\overrightarrow{AC}$ and $\overrightarrow{BC},$ which is $C.$

There is a rigid motion that maps $\triangle DEF$ onto $\triangle ABC.$ Consequently, $\triangle ABC$ and $\triangle DEF$ are indeed congruent.