Final answer:
The feathery arms of fanworms used for trapping food are called lophophores, which are specialized feeding structures consisting of ciliated tentacles around the mouth.
Step-by-step explanation:
The feathery arms of fanworms that trap food particles are called lophophores. Lophophores are feeding structures composed of a set of ciliated tentacles surrounding the mouth. These structures are characteristic of organisms in the superphylum Lophotrochozoa, which includes various species with either a lophophore feeding apparatus, like the fanworms, or trochophore larvae. In marine polychaetes (segmented worms), the lophophores serve as efficient means for filter-feeding.