Congruence of Triangles is Equivalence Relation

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Theorem

Let $S$ denote the set of all triangles in the plane.

Let $\triangle A \cong \triangle B$ denote the relation that $\triangle A$ is congruent to $\triangle B$.

Then $\cong$ is an equivalence relation on $S$.


Proof

Checking in turn each of the criteria for equivalence:


Reflexivity

Let $\triangle A$ be a triangle.

By definition, by Triangle Side-Side-Side Equality, $\triangle A$ is trivially congruent to itself.

Thus $\cong$ is seen to be reflexive.

$\Box$


Symmetry

Let $\triangle A \cong \triangle B$.


Then:

all the sides of $\triangle A$ are equal to the sides of $\triangle B$
all the angles contained by the sides of $\triangle A$ are equal to the angles contained by the sides of $\triangle B$.


It follows directly that:

all the sides of $\triangle B$ are equal to the sides of $\triangle A$
all the angles contained by the sides of $\triangle B$ are equal to the angles contained by the sides of $\triangle A$.


That is:

$\triangle B \cong \triangle A$

Thus $\cong$ is seen to be symmetric.

$\Box$


Transitivity

Let:

$\triangle A \cong \triangle B$
$\triangle B \cong \triangle C$


Then:

all the sides of $\triangle A$ are equal to the sides of $\triangle B$
all the angles contained by the sides of $\triangle A$ are equal to the angles contained by the sides of $\triangle B$.

and:

all the sides of $\triangle B$ are equal to the sides of $\triangle C$
all the angles contained by the sides of $\triangle B$ are equal to the angles contained by the sides of $\triangle C$.


From Equals is Equivalence Relation, it follows that:

all the sides of $\triangle A$ are equal to the sides of $\triangle C$
all the angles contained by the sides of $\triangle A$ are equal to the angles contained by the sides of $\triangle C$.


That is:

$\triangle A \cong \triangle C$

Thus $\cong$ is seen to be transitive.

$\Box$


$\cong$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.

$\blacksquare$


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