Angle-Side-Angle Congruence Theorem - Rule, Proof and Examples

This proof will be developed based on the given diagram, but it is valid for any pair of triangles.

The goal of the proof is to find a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways. One of the ways will be shown here.

Translate △ DEF So That Two Corresponding Vertices Match

Translate △ DEF so that D is mapped onto A. If this translation maps △ DEF onto △ ABC, the proof is complete.

Translation that maps vertex D of triangle DEF onto vertex A of triangle ABC

Since the image of the translation does not match △ ABC, at least one more transformation is needed.

Rotate △ AE'F' So That Two Corresponding Sides Match

Rotate △ AE'F' counterclockwise about A so that a pair of corresponding sides match. If the image of this transformation is △ ABC, the proof is complete. Note that this rotation maps E' onto B. Therefore, the rotation maps AE' onto AB.

Rotation that maps vertex E' of AE'F' onto B of ABC

As before, the image does not match △ ABC. Therefore, a third rigid motion is required.

Reflect △ ABF'' So That All Corresponding Sides Match

Reflect △ ABF'' across AB. Because reflections preserve angles, AF'' and BF'' are mapped onto AC and BC, respectively. Then, the point of intersection of the original rays F'' is mapped onto the point of intersection of the image rays C.

This time the image matches △ ABC.

Consequently, after a sequence of rigid motions △ DEF can be mapped onto △ ABC. This means that △ DEF and △ ABC are congruent triangles.

Proof

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