When constructing an angle bisector, the arcs must intersect because an angle bisector is a line or ray that divides an angle into two congruent angles. In order to divide the angle into two congruent angles, the angle bisector must meet at a common point within the angle. This point of intersection is the vertex of the angle. If the arcs do not intersect, it would mean that the angle bisector does not pass through the vertex of the angle and therefore, it would not be a true angle bisector.
