It could be either a birth defect - as Chief has mentioned - or an injury. If it's an injury, keep your water VERY clean and it will heal fast on its own. I'd recommend for the Nitrate not to exceed 10 ppm during the healing period. You keep Nitrate low by regular water changes. Ammonia and Nitrite should, of course, be at 0 ALWAYS
Chief, one of my Angels' "feeler" fin appeared to be half missing a while ago too. It grew back in less than a month. I don't think that's long. And whenever any of my Angels has a torn fin, it heals within 2 - 3 days. I believe all of this is due to keeping water clean.