Draw a ray with endpoint at the desired vertex.
Place the compass point at the vertex and draw an arc that cuts the ray at one point.
Without changing the compass width, place the compass point on the cut point and draw an arc inside the angle.
Mark a point on the arc where the second ray should pass.
Draw a ray from the vertex through the marked point.
The angle is constructed.
To construct a 60° angle, draw an arc from the vertex, then draw another arc from the cut point with the same compass width, and join the vertex to the intersection of the arcs.
To construct a 90° angle, construct a perpendicular to the ray using equal arcs from two points on the ray.
To construct a 45° angle, first construct a 90° angle and then bisect it.
To construct any angle, copy the angle using arcs and equal chord lengths.
To bisect an angle, draw an arc from the vertex cutting both sides of the angle, then draw equal arcs from those two cut points and join the vertex to their intersection.