An electronic device may be provided with antenna structures. The antenna structures may include a plate antenna. The electronic device may have a conductive housing such as a metal housing with an opening. A dielectric antenna window may be formed within the opening. A dielectric support structure such as a flexible printed circuit may overlap the opening. A conductive trace on the dielectric support structure may form an antenna resonating element plate for the plate antenna. The plate may have a periphery that is separated from adjacent portions of the metal housing by a gap. The antenna resonating element plate may have a rectangular shape with a bend that lies along an edge of the conductive housing. The dielectric antenna window may have a bend that also lies along the edge of the conductive housing.