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If the three sides of a triangle are congruent to the three sides of another triangle, then the triangles are congruent.
Based on the diagram above, the theorem can be written as follows.
AB ≅ DE BC ≅ EF AC ≅ DF ⇒ △ ABC ≅ △ DEF
The primary purpose of this 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 them will be shown here.
Since the image of the translation does not match △ ABC, at least one more transformation is needed.
As before, the image does not match △ ABC. Therefore, a third rigid motion is required.
It can be noted that AC = AF'' and BC = BF''. By the Converse Perpendicular Bisector Theorem, AB is a perpendicular bisector of CF''. Points along the perpendicular bisector are equidistant from the endpoints of the segment, so CG = GF''.
Finally, F'' can be mapped onto C by a reflection across AB by reflecting △ ABF'' across AB. Because reflections preserve angles, AF'' and BF'' are mapped onto AC and BC, respectively.
This time the image matches △ ABC.
Consequently, the application of a sequence of rigid motions allows △ DEF to be mapped onto △ ABC. This means that △ DEF and △ ABC are congruent triangles. The proof is complete.