# Reflexive Property of Equality

## Reflexive Property

The **reflexive property of equality** states that everything is equal to itself, whether it be a specific value or a mathematical expression. Examples of the reflexive property are shown below.

{eq}\hspace{2em} 3 = 3 {/eq}

{eq}\hspace{2em} -2\pi = -2\pi {/eq}

{eq}\hspace{2em} 5x^2 = 5x^2 {/eq}

{eq}\hspace{2em} 8a - b + \tfrac{1}{2}c = 8a - b + \tfrac{1}{2}c {/eq}

The name of the property comes from the word "reflection." That is because, in the reflexive property, the two sides of the equation are exact reflections of one another.

### Reflexive Property in Geometry

The reflexive property is often used in proofs in geometry. In geometry, the reflexive property of congruence states that any figure (line segment, angle, polygon, 3-dimensional shape, etc.) is congruent to itself. That is, every figure is the same size and shape as itself. Though congruence does not require a figure to be in the same orientation as a figure it is congruent to, obviously, every figure also has the same orientation as itself as well.

For instance, in the diagram below, the reflexive property can be used to say that {eq}\triangle ABC \cong \triangle ABC {/eq}. Here, the symbol {eq}\cong {/eq} means "is congruent to." So, {eq}\triangle ABC \cong \triangle ABC {/eq} means that triangle {eq}ABC {/eq} is congruent to itself. In the same diagram, the reflexive property could also be used to say that each line segment is congruent to itself (e.g. {eq}\overline{AB} \cong \overline{AB} {/eq}) or each angle is congruent to itself (e.g. {eq}\angle ABC \cong \angle ABC {/eq}).

There is also a reflexive property for similarity, which states that every figure is similar to itself. In geometry, similarity means that the two figures are the same shape but not necessarily the same size. So, for instance, by this property, one could write that {eq}\triangle ABC \sim \triangle ABC {/eq}, where the symbol {eq}\sim {/eq} means "is similar to."

### Importance of Reflexive Property

Although the reflexive property of equality is rarely used in algebraic manipulation, it can be used to prove other useful algebraic properties, like the addition property of equality or the multiplication property of equality. The reflexive property is more frequently used in geometry, where it is used to prove new theorems involving congruence and similarity.

## Reflexive Property Examples

### Reflexive Property of Equality Example 1

For instance, if it is given that {eq}a=b {/eq}, then the reflexive property can be used to prove that {eq}ac=bc {/eq}.

For this proof, start with the reflexive property itself, which states the following.

{eq}\hspace{2em} ac=ac {/eq}

Then since {eq}a=b {/eq}, it is possible to substitute {eq}b {/eq} for {eq}a {/eq} in the equation above.

{eq}\hspace{2em} ac=bc {/eq}

This theorem (that is, the idea that if {eq}a=b {/eq}, then {eq}ac=bc {/eq} must be true as well) is called the *multiplication property of equality*.

A similar proof involving the reflexive property of equality can be used to prove the addition property of equality, which states that if {eq}a=b {/eq}, then {eq}a+c=b+c {/eq} must be true. It can also be used to prove the subtraction property and the division property of equality.

### Reflexive Property of Equality Example 2

In this geometric example, assume that {eq}\overline{AB} \cong \overline{AC} {/eq} and {eq}\overline{BM} \cong \overline{MC} {/eq} in the diagram below, as shown by the hatch marks on these line segments. Using these facts, it can be proved that the two triangles are congruent to one another as well.

The key to this proof is to use the reflexive property, which states that {eq}\overline{AM} \cong \overline{AM} {/eq}. That is, the middle segment is congruent to itself. If this is combined with the two given facts ({eq}\overline{AB} \cong \overline{AC} {/eq} and {eq}\overline{BM} \cong \overline{MC} {/eq}), this means that the three sides of the {eq}\triangle ABM {/eq} are congruent to the three sides of {eq}\triangle AMC {/eq}. Therefore, by the SSS congruence theorem of triangles, it can be concluded that the two triangles are congruent to one another i.e. {eq}\triangle ABM \cong \triangle ACM {/eq}.

## Lesson Summary

The **reflexive property of equality** states that everything is equal to itself, whether it be a specific value or a mathematical expression.

In geometry, the reflexive property of congruence states that any figure (line segment, angle, polygon, 3-dimensional shape, etc.) is congruent ({eq}\cong {/eq}) to itself. That is, every figure is the same size and shape as itself. The reflexive property of similarity ({eq}\sim {/eq}) states an analogous thing for similarity (i.e. figures that are the same shape but not necessarily the same size).

Examples of the reflexive property are the equation {eq}3 = 3 {/eq} as well as {eq}\overline{AB} \cong \overline{AB} {/eq} and {eq}\triangle XYZ \sim \triangle XYZ {/eq}.

Although the reflexive property of equality is rarely used in algebraic manipulation, it can be used to prove other useful algebraic properties, like the addition property of equality or the multiplication property of equality. The reflexive property is more frequently used in geometry, where it is used to prove new theorems involving congruence.

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#### What are reflexive relations examples?

Some examples of reflexive relations are the equations 3 = 3, -0.21 = -0.21, 5x^2 = 5x^2, and 8a-b+1/2c = 8a-b+1/2c.

#### What is a reflexive property?

The reflexive property of equality states that everything is equal to itself, whether it be a specific value or a mathematical expression.

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