Congruent Polygons

1. If ABCD and PQRS are congruent, then their corresponding sides and angles are congruent.
2. If the corresponding sides and angles of ABCD and PQRS are congruent, then the polygons are congruent.

Part 1

By the definition of congruent figures, if the polygons are congruent there is a rigid motion or sequence of rigid motions that maps ABCD onto PQRS.

Because rigid motions preserve side lengths, AB and its image have the same length — that is, AB=PQ. Therefore, AB and PQ are congruent segments. Similar observations are true for the other three sides. BC≅QR CD≅RS AD≅PS Furthermore, rigid motions preserve angle measures, which means that ∠ A and its image have the same measure. Since m∠ A = m∠ P, ∠ A and ∠ P are congruent angles. Similarly, all the remaining angles can also be concluded to be congruent. ∠ B ≅ ∠ Q ∠ C ≅ ∠ R ∠ D ≅ ∠ S In this fashion, it has been shown that if two polygons are congruent, then their corresponding sides and angles are congruent.

Part 2

To begin, congruent parts on the given diagram will be marked.

The primary purpose of this part is to find a rigid motion or sequence of rigid motions that maps one polygon onto the other. This can be done in several ways, and what is shown here is only one.

1

Translate ABCD So That One Pair of Corresponding Vertices Match

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Apply a translation that maps D onto S. If this translation maps ABCD onto PQRS, the proof will be complete.

As can be seen, A'B'C'D' did not map onto PQRS. Therefore, a second rigid motion is needed.

2

Rotate A'B'C'S So That One Pair of Corresponding Sides Match

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Apply a clockwise rotation about S through ∠ RSC' to A'B'C'S. If the image matches PQRS, the proof will be complete. Note that this rotation maps C' onto R and therefore SC' onto SR.

The image still does not match PQRS, so a third rigid motion is required.

3

Reflect A''B''RS So That the Corresponding Sides Match

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Finally, apply a reflection across RS to A''B''RS. Because reflections preserve angles and lengths, SA'' is mapped onto SP and RB'' is mapped onto RQ. Likewise, A''B'' is mapped onto PQ.

This time the image matches PQRS.

Consequently, through applying a series of different rigid motions, ABCD was mapped onto PQRS. This implies that ABCD and PQRS are congruent polygons. With this, the proof is complete.