The "only if" direction of this proof is easy to show, since the congruence of two arcs means by definition that their angles are congruent.
The "if" direction requires you to show that their arc lengths are equal. Recall the formula for arclength: $2\pi r\frac{\theta}{2\pi}$ where $r$ is the radius of the circle containing the arc, and $\theta$ is the central angle of the arc. If the first arc belongs to circle $C_1$ with radius $r_1$ and the second belongs to circle $C_2$ with radius $r_2$, and their central angles are congruent, then you need only show that $r_1=r_2$ to get equality of the arc lengths, and hence, congruence of the arcs. If they belong to the same circle, this is obvious. If they belong to congruent circles, then again, by definition of congruence, their radii must be equal.