We compute the capacity region of a sum of broadcast channels whose components are degraded, less-noisy, more-capable, deterministic, or semi-deterministic. We achieve this by showing that an auxiliary-receiver outer bound, previously introduced by some of the authors, matches Marton's inner bound. This result generalizes a previously known result for the sum of two reversely degraded broadcast channels due to El Gamal (1980). Moreover, we define a class of primary broadcast channels and show an analogous result for the sum of primary broadcast channels.