Reference

Properties of Congruence

Rule

Reflexive Property of Congruence

Any geometric figure is congruent to itself. For example, a segment $\overline{A B}$ is always congruent to itself.

$\overline{A B} ≅ \overline{A B}$

Proof

This proof can be demonstrated using any geometric figure. For simplicity, the quadrilateral $A B C D$ shown below will be considered.

By definition, two figures are congruent if and only if they have the same size and shape. Also, congruent figures can be placed onto each other by a combination of rigid motions. Therefore, it should be investigated whether

A B C D

can be placed onto itself by using one or a combination of rigid motions.

The diagram illustrates how

A B C D

places onto itself after a rotation of

36 0^{\circ}

about

D .

Therefore, since a rotation is a rigid motion, it can be concluded that

A B C D

is congruent to itself.

$A B C D ≅ A B C D$

The proof is complete.

Rule

Symmetric Property of Congruence

If a given geometric object $A$ is congruent to another geometric object $B,$ then $B$ is also congruent to $A .$

If $A ≅ B,$ then $B ≅ A .$

Proof

This proof can be demonstrated using any geometric figure. For simplicity, the triangles $△ A B C$ and $△ D E F$ will be considered.

By definition, two figures are congruent if and only if they have the same size and shape. Additionally, congruent figures can be placed onto each other by a combination of rigid motions. Since

△ A B C

is congruent to

△ D E F,

there is a combination of rigid motions that places

△ A B C

onto

△ D E F .

From the diagram, it can be seen that the combination of a rotation and a translation places

△ A B C

onto

△ D E F .

Using a similar combination of rigid motions,

△ D E F

can also be placed onto

△ A B C .

Therefore,

△ D E F

is congruent to

△ A B C .

$△ D E F ≅ △ A B C$

The proof is now complete.

Rule

Transitive Property of Congruence

If two geometric objects $A$ and $B$ are congruent, and $B$ is also congruent to $C,$ then $A$ and $C$ are congruent.

If $A ≅ B$ and $B ≅ C,$ then $A ≅ C .$

Proof

This proof can be demonstrated using any geometric figure. For simplicity, the polygons $A,$ $B,$ and $C$ will be considered.

By definition, two figures are congruent if and only if they have the same size and shape. Also, congruent figures can be placed onto each other by a combination of rigid motions. Since

A

and

B

are congruent figures, there is a combination of rigid motions that place

A

onto

B .

In this case, the combination of a rotation and a translation places

A

onto

B .

Similarly, since

B

and

C

are congruent figures, there is a combination of rigid motions that places

B

onto

C .

The diagram illustrates with a rotation and a translation,

B

places onto

C .

Collectively, the rigid motions that place

A

onto

B

and

B

onto

C

are a combination of rigid motions. Consequently, this combination of rigid motions places

A

onto

C .

Therefore,

A

and

C

are congruent figures.

$A ≅ C$

The proof is now complete.

Proof

Proof

Proof

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