The following are the properties of congruence. Some textbooks list just a few of them, others list them all. These are analogous to the properties of equality for real numbers. Here we show congruences of angles, but the properties apply just as well for congruent segments, triangles, or any other geometric object.
PROPERTIES
OF CONGRUENCE 

Reflexive Property  For all angles $A$, $\angle A\cong \angle A$ . An angle is congruent to itself. 
These
three properties define an equivalence relation

Symmetric Property  For any angles $A\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}B$, if $\angle A\cong \angle B$, then $\angle B\cong \angle A$. Order of congruence does not matter. 

Transitive Property  For any angles $A,B,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}C$, if $\angle A\cong \angle B$ and $\angle B\cong \angle C$, then $\angle A\cong \angle C$. If two angles are both congruent to a third angle, then the first two angles are also congruent. 