your gander's lucky the fox didn't have any experience hunting geese or he would of been go with the goose. Geese do little to protect your birds I can say they will help prevent some hawk attacks. owls that come at night will take what they can get. AS for the golden eagle they eat full grown cranes, will drag baby sheep off cliffs, and falconers use them to hunt wolves, so the geese could fall pray to the eagle, I'm not sure about falcons it depends on how big and hungry it is. As for badgers as long as your chickens can get off the ground or should I say are smart enough to get off the ground they should be safe. Coyotes and fox will take both the chickens and geese if they can. Your biggest problem might come from the small weasels, but if you know how to trap them they shouldn't be a problem. So geese may do a little to help but don't depend on them fully, a dog that is bigger then 50 pounds would keep the fox, coyote, badger, may catch the weasels, and could help with the birds of pray.