The study of EpGs (extended partial geometries) is continued from [5]; here we consider extended dual 2-designs (EDDs), especially one-point and triangular structures. We concentrate on three interesting cases: semibiplanes, which are extensions of the duals of trivial 2-designs (Section 4); extensions of dual projective geometries (Section 7); extensions of dual affine geometries (Section 8). We find non-existence theorems, examples, uniqueness theorems, and many open questions