If you look closely, every great circle and every small circle on a stereonet intersect at right angles, just like on the sphere. Therefore, the angle between the planes is simply the angle at which their great circles intersect on the stereonet. To measure the angle precisely, retrace the great circles as exactly as possible near the intersection and construct tangents to the arcs. Measure the angles between the tangents. And it's a nice illustration of the conformality property.
Some geologists prefer to work with the equal area projection.
This method will not work in that case, although the pole method below works for any projection. To measure the angle between planes, we have to make a cross-section looking down their intersection. The cross-section is perpendicular to both planes, therefore it contains the poles. It's easy to see that the angle between the planes is equal to the angle between the poles . The same problem as above is shown below, only using the poles of the planes.