When Dogs Attack

We had an incidence lately, reported in the local paper, of two neighborhood dogs attacking and killing livestock. The dogs were a Pit Bull and a Great Dane, and they killed two sheep and one llama, and injured a third sheep so badly it had to be put down. They were running loose in their own neighborhood, and apprehended because they ended up in another person's house and charged her when she tried to get them to leave. Their guilt was fairly well proven because they both had blood from the livestock on them.
The Great Dane was a repeat offender, having killed a llama near his previous home in the next town over. The owner had him put down by the local humane society, which was holding both dogs. The fate of the Pit Bull has not yet been decided. The owner has the option of keeping the dog, but as a known "dangerous dog," with a plethora of legal restrictions. The dog will have to be kept in a kennel complete with floor and ceiling and will have to wear a muzzle whenever out of the kennel. The owner will have to carry liability insurance for the dog.
We live in a fairly rural area, and livestock is common. When we first moved here, before even acquiring sheep, a local free-roaming pack visited our property several times. Each time I went out after them, whooping and waving a stout stick. The apparent pack leader, a Siberian Husky or mix thereof, appeared to consider challenging me, but decided against it. After the third chase, I never saw them again. I don't know if the husky was hit by a car (a common occurrence with loose-running dogs, was surrendered, met some other fate, or just decided to visit someone less serious about defending property. But by the time we did have sheep and chickens, we were not being visted by any dogs.
Would I shoot a dog that was attacking my livestock? Possibly. The sheep and llamas are my pets, too, not just wool-bearing creatures. And it would be legal. But could I really pull the trigger on a dog? I'm just not sure. I hope I never have to make the choice.