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 ABC\triangle ABC and DEF,\triangle DEF, where AD,ABDE, and  BE. \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 AB\overline{AB} is congruent with DE,\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.


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

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

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

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