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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 then

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 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.

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.

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 and are congruent, then their corresponding sides and angles are congruent.
  2. If the corresponding sides and angles of and are congruent, then the triangles are congruent.

Part

By definition of congruent figures, if the triangles are congruent there is a rigid motion or sequence of rigid motions that maps onto
Mapping ABC onto DEF
Because rigid motions preserve side lengths, and its image have the same length, that is, Therefore, Similarly for the other two side lengths.
Furthermore, rigid motions preserve angle measures. Then, and its image have the same measure, that is, Therefore, Similarly for the remaining angles.
That way, it has been shown that if two triangles are congruent, then their corresponding sides and angles are congruent.

Part

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 so that one pair of corresponding vertices match
expand_more
Apply a translation to that maps to If this translation maps onto the proof will be complete.
Translating Triangle ABC
As seen, did not match Therefore, a second rigid motion is needed.
2
Rotate so that one pair of corresponding sides match
expand_more
Apply a clockwise rotation to about through If the image matches the proof will be complete. Notice this rotation maps onto and therefore, onto
Rotating Triangle DB'C'
As before, the image did not match Thus, a third rigid motion is required.
3
Reflect so that the corresponding sides match
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Apply a reflection to across Because reflections preserve angles, is mapped onto and is mapped onto Then, the intersection of the original rays is mapped to the intersection of the image rays
Reflecting Triangle DEC''
This time the image matched

Consequently, through applying different rigid motions, was mapped onto This implies that and 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.

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

Part

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

Polygons ABCD and PQRS Rigid Motion
Because rigid motions preserve side lengths, and its image have the same length — that is, Therefore, and are congruent segments. Similar observations are true for the other three sides.
Furthermore, rigid motions preserve angle measures, which means that and its image have the same measure. Since and are congruent angles. Similarly, all the remaining angles can also be concluded to be congruent.
In this fashion, it has been shown that if two polygons are congruent, then their corresponding sides and angles are congruent.

Part

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 So That One Pair of Corresponding Vertices Match
expand_more
Apply a translation that maps onto If this translation maps onto the proof will be complete.
Translating Polygon ABCD
As can be seen, did not map onto Therefore, a second rigid motion is needed.
2
Rotate So That One Pair of Corresponding Sides Match
expand_more
Apply a clockwise rotation about through to If the image matches the proof will be complete. Note that this rotation maps onto and therefore onto
Rotation Polygon ABCD
The image still does not match so a third rigid motion is required.
3
Reflect So That the Corresponding Sides Match
expand_more
Finally, apply a reflection across to Because reflections preserve angles and lengths, is mapped onto and is mapped onto Likewise, is mapped onto
Reflecting Polygon ABCD
This time the image matches

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