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Reference

Properties of Congruence

Rule

Reflexive Property of Congruence

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

AB ≅AB

Proof
This proof can be demonstrated using any geometric figure. For simplicity, the quadrilateral ABCD 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 ABCD 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 ABCD places onto itself after a rotation of 360^(∘) about D. Therefore, since a rotation is a rigid motion, it can be concluded that ABCD is congruent to itself.

ABCD ≅ ABCD

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 â–³ ABC and â–³ DEF 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 â–³ ABC is congruent to â–³ DEF, there is a combination of rigid motions that places â–³ ABC onto â–³ DEF.

Triangle ABC maps onto triangle DEF

From the diagram, it can be seen that the combination of a rotation and a translation places â–³ ABC onto â–³ DEF. Using a similar combination of rigid motions, â–³ DEF can also be placed onto â–³ ABC.

Triangle DEF places onto triangle ABC

Therefore, â–³ DEF is congruent to â–³ ABC.

△ DEF≅ △ ABC

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.

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 A and B are congruent figures, there is a combination of rigid motions that place A onto B.

A maps 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.

B maps 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.
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