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Two figures are congruent figures if there is a rigid motion or sequence of rigid motions that maps one of the figures onto the other. As a result, congruent figures have the same size and shape. To denote algebraically that two figures are congruent, the symbol

$≅$is used.

When writing a polygon congruence, the corresponding vertices must be listed in the same order. For the polygons above, two of the possible congruence statements can be written as follows.

$ABCDE≅JKLMNorCDEAB≅LMNJK $

Two triangles are congruent if and only if their corresponding sides and angles are congruent.

Using the triangles shown, this claim can be written algebraically as follows.

$△ABC≅△DEF⇕AB≅DEBC≅EFAC≅DF and∠A≅∠D∠B≅∠E∠C≅∠F $

This proof will be developed based on the given diagram, but it is valid for any pair of triangles. The proof of this biconditional statement consists of two parts, one for each direction.

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

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≅EFandAC≅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≅∠Eand∠C≅∠F $
That way, it has been shown that if two triangles are congruent, then their corresponding sides and angles are congruent.

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.

Translate $△ABC$ so that one pair of corresponding vertices match

As seen, $△A_{′}B_{′}C_{′}$ did not match $△DEF.$ Therefore, a second rigid motion is needed.

Rotate $△DB_{′}C_{′}$ so that one pair of corresponding sides match

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

Reflect $△DEC_{′′}$ so that the corresponding sides match

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.

Two polygons are congruent if and only if their corresponding sides and angles are congruent.

Using the polygons shown, this claim can be written algebraically as follows.

$ABCD≅PQRS⇕ABBCCDAD ≅PQ ≅QR ≅RS≅PS and∠A≅∠B≅∠C≅∠D≅ ∠P∠Q∠R∠S $

This proof will be developed based on the given diagram, but it is valid for any pair of polygons. The proof of this biconditional statement consists of two parts, one for each direction.

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

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≅RSAD≅PS $
Furthermore, rigid motions preserve angle measures. Then, $∠A$ and its image have the same measure, that is, $m∠A=m∠P.$ Therefore, $∠A$ and $∠P$ are congruent angles. Similarly, all the remaining angles can also be concluded to be congruent.
$∠B≅∠Q∠C≅∠R∠D≅∠S $
That way, it has been shown that if two polygons are congruent, then their corresponding sides and angles are congruent.

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 polygon onto the other. This can be done in several ways, here is only one of them shown.Translate $ABCD$ So That One Pair of Corresponding Vertices Match

As seen, $A_{′}B_{′}C_{′}D_{′}$ did not map onto $PQRS.$ Therefore, a second rigid motion is needed.

Rotate $A_{′}B_{′}C_{′}S$ So That One Pair of Corresponding Sides Match

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

Reflect $A_{′′}B_{′′}RS$ So That the Corresponding Sides Match

This time the image matched $PQRS.$

Consequently, through applying different rigid motions, $ABCD$ was mapped onto $PQRS.$ This implies that $ABCD$ and $PQRS$ are congruent polygons. Hence, the proof is complete.

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