Congruent Triangles

1. If △ ABC and △ DEF are congruent, then their corresponding sides and angles are congruent.
2. If the corresponding sides and angles of △ ABC and △ DEF are congruent, then the triangles are congruent.

Part 1

By definition of congruent figures, if the triangles are congruent there is a rigid motion or sequence of rigid motions that maps △ ABC onto △ DEF.

Because rigid motions preserve side lengths, AB and its image have the same length, that is, AB=DE. Therefore, AB≅DE. Similarly for the other two side lengths. BC≅EF and AC≅DF Furthermore, rigid motions preserve angle measures. Then, ∠ A and its image have the same measure, that is, m∠ A = m∠ D. Therefore, ∠ A ≅ ∠ D. Similarly for the remaining angles. ∠ B ≅ ∠ E and ∠ C ≅ ∠ F That way, it has been shown that if two triangles are congruent, then their corresponding sides and angles are congruent.

Part 2

To begin, mark the congruent parts on the given diagram.

The primary purpose is finding a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways, here it is shown one of them.

1

Translate △ ABC so that one pair of corresponding vertices match

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Apply a translation to △ ABC that maps A to D. If this translation maps △ ABC onto △ DEF the proof will be complete.

As seen, △ A'B'C' did not match △ DEF. Therefore, a second rigid motion is needed.

2

Rotate △ DB'C' so that one pair of corresponding sides match

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Apply a clockwise rotation to △ DB'C' about D through ∠ EDB'. If the image matches △ DEF, the proof will be complete. Notice this rotation maps B' onto E, and therefore, DB' onto DE.

As before, the image did not match △ DEF. Thus, a third rigid motion is required.

3

Reflect △ DEC'' so that the corresponding sides match

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Apply a reflection to △ DEC'' across DE. Because reflections preserve angles, DC'' is mapped onto DF and EC'' is mapped onto EF. Then, the intersection of the original rays C'', is mapped to the intersection of the image rays F.

This time the image matched △ DEF.

Consequently, through applying different rigid motions, △ ABC was mapped onto △ DEF. This implies that △ ABC and △ DEF are congruent. Then, the proof is complete.