Its sad, but many times its the only solution.
Some years people dump so many dogs in my part of the country that I'll have to dispose of a dozen or more. They'll form packs and run cattle to death at times.
Sorry you had to, but it has to be done sometimes :(
That is crazy - a dozen or more?!? Do you have any advice on how to actually get them since you deal with them so much? I know there are more out here, too. I talked to one neighbor who said that there was someone down the road who has a big dog that they don't feed very well and just let it "forage" for food. Not sure exactly what they expect a hungry dog to do... Especially because it is not locked up!
Getting a large livestock guardian dog is a good, yet expensive start.
Practice safe shooting skills until you're able to put 3 rounds in a 3 inch circle at 100 meters with soft or hollow-point hunting rounds. While unpleasant, it will help to ensure a quick, humane kill.
Post "No Trespassing" and "Trapping" signs on your property boundaries, but DONT SET TRAPS and don't tell anyone you don't set traps. While predators can't read them, dogs owners can.
Talk with your neighbors and politely let them know that whatever happens, its not personal, you're just protecting your property.
I know it sounds harsh, but it works for me.