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Reference

Defining Congruence for Different Mathematical Objects

Concept

Congruent Angles

Two angles are congruent angles if both have the same measure. In a diagram, congruent angles are usually indicated by the same number of angle markers.

Two congruent angles

The symbol ≅ is used to algebraically express that two angles are congruent.
Concept

Congruent Segments

Two segments are congruent segments if both have the same length. In a diagram, congruent segments are usually indicated by the same number of ticks. The symbol ≅ is used to express algebraically that two segments are congruent.

Two congruent segments AB and CD

Therefore, if AB=CD, then AB≅CD.
Concept

Congruent Figures

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.

Showing that two figures are congruent by mapping one onto the other
When stating that two polygons are congruent, 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 ≅ JKLMN or CDEAB ≅ LMNJK

Rule

Congruent Triangles

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

Triangles ABC and DEF

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

△ ABC ≅ △ DEF ⇕ AB≅DE BC≅EF AC≅DF and ∠A≅∠D ∠B≅∠E ∠C≅∠F

Proof
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.

  1. If â–³ ABC and â–³ DEF are congruent, then their corresponding sides and angles are congruent.
  2. If the corresponding sides and angles of â–³ ABC and â–³ 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 â–³ ABC onto â–³ DEF.

Mapping 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.

Marking the congruent parts

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 â–³ ABC so that one pair of corresponding vertices match
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Apply a translation to â–³ ABC that maps A to D. If this translation maps â–³ ABC onto â–³ DEF the proof will be complete.

Translating Triangle ABC

As seen, â–³ A'B'C' did not match â–³ DEF. Therefore, a second rigid motion is needed.

2
Rotate â–³ 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.

Rotating Triangle DB'C'

As before, the image did not match â–³ DEF. Thus, a third rigid motion is required.

3
Reflect â–³ DEC'' so that the corresponding sides match
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Apply a reflection to â–³ 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.

Reflecting Triangle DEC''

This time the image matched â–³ DEF.



Consequently, through applying different rigid motions, â–³ ABC was mapped onto â–³ DEF. This implies that â–³ ABC and â–³ DEF are congruent. Then, the proof is complete.
Rule

Congruent Polygons

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

Polygons ABCD and PQRS

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

ABCD ≅ PQRS ⇕ AB&≅PQ BC&≅QR CD&≅RS AD&≅PS and ∠A≅&∠P ∠B≅&∠Q ∠C≅&∠R ∠D≅&∠S

Proof

 Proving Congruence in Polygons
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.

  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.

Polygons ABCD and PQRS Rigid Motion

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.

Polygons ABCD and PQRS

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.

Translating Polygon ABCD

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.

Rotation Polygon ABCD

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.

Reflecting Polygon ABCD

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.
Exercises

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