No, you can't add your betta to your tank with the angels and the goldfish. The angels would rip the poor betta to pieces (remember, angels are cichlids and can be territorial and aggressive).
I would just like to point out that your angels and goldfish are incompatible, as well. Angelfish are tropical fish and prefer temperatures over 78F. Goldfish are temperate, or "coldwater," fish and prefer temperatures below 75F. Also, goldfish are HUGE waste producers and grow to be quite large, so the general recommendation is to have 20 gallons for the first goldfish and 10-20 gallons per fish after that. So, with just the one goldfish, you're tank is already stocked. Goldfish also need a tank that is overfiltered by at least double because of the amount of waste they produce (with just the one male ryukin goldfish in my 46gal bowfront, I am running two filters: a Penguin 170 rated for up to 40 gallons and an AquaTech 30-60 rated for up to 60 gallons...a canister filter is much preferred in goldfish tanks, I just can't afford one right now).
To add to the above issues, the angelfish should really be in a larger tank. They get to be 6" long and 10" tall and a 20gal tank just does not provide enough swimming room for them. I generally recommend keeping angelfish in nothing smaller than a 35-40gal tank, with taller being better than wider.
About your betta in the bowl...bettas require filtered, heated, cycled tanks of a minimum 2.5 gallons to thrive (I prefer larger, personally...my boys are in nothing smaller than a 6gal). Your comment about your betta's bowl "never [neeeding] cleaning" seriously made me shudder. Do you understand the nitrogen cycle? Basically, your fish poops, which creates ammonia, as does any decaying organic matter, such as leftover food. Ammonia is INCREDIBLY toxic to fish. So, the presence of ammonia jump-starts the life cycle of a beneficial bacteria that converts the toxic ammonia into slightly-less-toxic nitrite. The presence of the nitrite jump-starts the life cycle of another beneficial bacteria that converts the still-toxic nitrite into relatively-harmless nitrate (which is only harmful in large quantities). In a 1 gallon bowl, you would need to clean it daily to keep the ammonia levels at zero (where they cannot hurt your fish), because there is no filter for the beneficial bacteria to establish on and work its magic rendering the ammonia and nitrites harmless. Also, bettas are tropical fish (like angelfish are) and require warm water. My betta tanks stay anywhere from 78-80F and my boys LOVE it. They are active and fat and healthy. Bettas do not come from mud puddles or rice paddies, as most people for some reason believe. They come from tropical pools where the water is warm and clean. The only time they are in really shallow water (under 5") is during the dry season, which only lasts a couple of months. Get your betta boy a 5 gallon tank that is heated, filtered and cycled and you'll find out what a truly happy betta is.
