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Reference

Properties of Congruence

Rule

Reflexive Property of Congruence

Any geometric figure is congruent to itself. For example, a segment is always congruent to itself.

Proof

This proof can be demonstrated using any geometric figure. For simplicity, the quadrilateral shown below will be considered.

A quadrilateral $ABCD$
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 can be placed onto itself by using one or a combination of rigid motions.
After rotating, the quadrilateral places onto itself
The diagram illustrates how places onto itself after a rotation of about Therefore, since a rotation is a rigid motion, it can be concluded that is congruent to itself.

The proof is complete.

Rule

Symmetric Property of Congruence

If a given geometric object is congruent to another geometric object then is also congruent to

If then

Proof

This proof can be demonstrated using any geometric figure. For simplicity, the triangles and will be considered.

Two triangles ABC and DEF
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 is congruent to there is a combination of rigid motions that places onto
Triangle ABC maps onto triangle DEF
From the diagram, it can be seen that the combination of a rotation and a translation places onto Using a similar combination of rigid motions, can also be placed onto
Triangle DEF places onto triangle ABC
Therefore, is congruent to

The proof is now complete.

Rule

Transitive Property of Congruence

If two geometric objects and are congruent, and is also congruent to then and are congruent.

If and then

Proof

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

Three polygons A, B, and C
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 and are congruent figures, there is a combination of rigid motions that place onto
A maps onto B
In this case, the combination of a rotation and a translation places onto Similarly, since and are congruent figures, there is a combination of rigid motions that places onto
B maps onto C
The diagram illustrates with a rotation and a translation, places onto Collectively, the rigid motions that place onto and onto are a combination of rigid motions. Consequently, this combination of rigid motions places onto Therefore, and are congruent figures.

The proof is now complete.

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