When two lines intersect, they form two pairs of vertical angles. Pairs of non-adjacent angles with common vertex formed by two intersecting lines are congruent.
In a circle, when two lines intersect at the center of the circle, the vertical angles formed are central angles. The measure of the central angle is equal to degree measure of the arc it intercepted.
So if vertical angles are congruent, then the arcs they intercept are also congruent.
Use the Vertical angle theorem, Central angle and Arc postulates in proving :-)