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Cooper's hawks in the west tend to eat more rodents than birds than they do in other parts of the country. We have several at the lake I go to and they never bother the adult ducks there, though the ducks are wary of them (they look like falcons, which are duck eaters and that could be why). However, they will eat a duckling up to about 2 to 3 weeks old. Cooper's hawks are not really strong and can't carry a lot of weight, so when they kill something they either have to eat it where it lies or carry it away to a tree. It would rather carry it to a tree and I know ducks are just too big for it to do that. I'm pretty sure most adult chickens are, too.
Besides rodents, most of the Cooper's hawks I see hunt medium-sized birds like grackles, doves, pigeons, starlings, and sometimes killdeer. I haven't seen them try for anything larger than a squirrel. That won't mean that they might not try to get a chicken or kill a chicken and abandon it, I just haven't heard of it doing so. Now red-tails on the other hand. . . .