In the construction of a triangle, the vertices are determined by:

The vertices of a triangle are primarily determined by the intersections of arcs when using compass and straightedge construction methods. In geometric constructions, you usually start by drawing a segment or an arc from a specific point (often one of the triangle's vertices). By drawing additional arcs from other points, the points where these arcs intersect will define the triangle's vertices.

For example, if you know the lengths of the sides of the triangle, you can construct arcs with the compass from each vertex to find the other vertices. The intersections of these arcs will give you the exact points where the vertices of the triangle are located. This method illustrates that the intersections of the arcs are critical in accurately determining the vertices, which is especially useful in a construction context.

Other options, while they relate to triangle properties, do not define the vertices directly. The lengths of the sides will describe the triangle's dimensions, but they do not point out where the vertices reside without additional context. Similarly, knowing the angles at each vertex or having one side as a base provides valuable information about the triangle but does not specify the exact vertex locations without performing a construction technique that leads to those intersections.