What you want to do is to dimension a vertex.
Start the dimangular command, then right click. The prompt now changes to "specify angle vertex". Choose the vertex with the endpoint snap, then you have to specify the first angle endpoint, and then the second angle endpoint. Then, all depending which side of the vertex is the crosshair, you can dimension the angle greater than 180° (you can move the crosshair each side of the vertex, then pick which one you want).
Not very intuitive, but when you have done it once, you get the hang of it!