Question

When constructing an angle bisector, why must the arcs intersect ?

Answer 1

The intersection of the two arcs are being used to show a point that is equidistant from the respective endpoints of your angle. Given that you have an angle, and use a compass set at a specific width, if you trace a circle around a single endpoint of the angle, you have a circle centered around that endpoint with a specific radius. Now if you do the same for the other endpoint of the angle, you have an identical circle, but this time centered on the newest endpoint. Since everywhere on the circle is the same distance (the width of the compass) from their respective endpoints, it can be inferred that the two points where these circles intersect are equidistant to both endpoints. This means that the intersections lie on the line that represents all the points equidistant to the two endpoints, i.e. the line that cuts the distance (or angle) between then in half.