Congruence Is Reflexive, Symmetric, and Transitive

First let's review what reflexive, symmetric & transitive means.

After reviewing these properties, let's see if we can apply them to congruence. Recall congruence means two objects are physically identical. In the world of geometry, they have the same shape and size.

Reflexive $\\$ Think of an object--it can be an object you see on your desk, in your room; or it can be a geometric figure: a square, a triangle, etc. Is this object congruent to itself? Is a mug physically identical to itself? Does a triangle have the same shape and size as itself? $\\$

The answer is yes to all these questions. A geometric figure is always congruent to itself. This means congruence is reflexive.

Symmetric $\\$ Now imagine the following scenario. You recently saw your friend wearing a pair of awesome sneakers. Knowing you have the same size feet as your friend, you asked what his/her size is. You then went online and found the same pair of sneakers and purchased it. The pair of new sneakers you got are, evidently, identical--or congruent to your friend's sneakers. So, are your friend's sneakers also congruent to yours? $\\$ Taking this a step further, if a geometric figure $A$ is congruent to another geometric figure $B$, is $B$ also congruent to $A$? $\\$

The answer is yes to both questions. If $A$ is congruent to $B$, $B$ is also congruent to $A$. This means congruence is symmetric.

Transitive $\\$ You recently bought an agenda. Your friend came over to hang out one afternoon and commented, "Hey! I got the exact same agenda. You know what though, Brenda said the other day her new agenda was the same as the one i got." Can you say you got the same agenda as Brenda?

Furthermore, say we have $\triangle ABC, \triangle DEF and \triangle GHI$. If $\triangle ABC \cong \triangle DEF$, $\triangle DEF \cong \triangle GHI$, can we say $\triangle ABC \cong \triangle GHI$? Does this apply to all geometric figures?

In fact, the answer is yes to all the questions above. If $A$ is congruent to $B$, $B$ is congruent to $C$, then $A$ is congruent to $C$. This means congruence is transitive.

Conclusion $\\$ Congruence is reflexive, symmetric and transitive.