Angles are congruent if they have the same angle measure. Two angles are congruent if they have the same measure in degrees.

Its not necessary for the angles to have same direction or equal segment lengths. But if the congruent angles are laid on top of each other they coincide with each other.

Congruent Angles

Geometric figures are classified based on the interior angles:

For Example:
Triangles: if all the interior angles of a triangle are congruent them it is called as an equilateral triangle.

Quadrilateral: If all the interior angles of a quadrilateral are congruent then it can be a square or a rectangle depending on the length of its sides.

Polygon: A polygon with all congruent interior angles is called as a regular polygon.

Here are some examples of congruent interior angles.
Congruent Interior Angles
Congruent Interior Angles

The following list gives some useful properties of the congruence of angles.

Reflexive: $\angle$ 1 = $\angle$1; an angle is congruent to itself

Symmetric: If $\angle$1 = $\angle$2 then $\angle$ 2 = $\angle$1.

Transitive: If $\angle$1 = $\angle$2 and $\angle$ 2 = $\angle$3 then $\angle$1 = $\angle$ 3.