Angle-Angle-Side Congruence Theorem

The primary purpose of the proof is finding 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.

1

Translate △ DEF So That Two Corresponding Vertices Match

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

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

2

Rotate △ CD'E' So That Two Corresponding Sides Match

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Rotate △ CD'E' clockwise about C 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 CE' onto CB.

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

3

Reflect △ ABF'' So That Two More Corresponding Sides Match

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It is given that two angles of △ ABC are congruent to two angles of △ BCD''. Hence, by the Third Angle Theorem, ∠ BCD'' is congruent to ∠ BCA.

Reflect △ CBD'' across BC. Because reflections preserve angles, BD'' and CD'' are mapped onto BA and CA, respectively. Then, the point of intersection of the original segments D'' is mapped onto the point of intersection of the image segments A.

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. The proof is complete.