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Yes. If you see your neighbor's dog harassing your livestock, by chasing them or worrying them, you are justified in killing the dog with no liability to your neighbor, so long as your livestock are on your property when they are being harassed. You may also notify the neighbor about their dog's behavior and they will have 24 hours to kill their dog after you have notified them, and if they do not kill their dog you can notify a law enforcement officer who will kill the dog.
This rule does not apply to dogs that are being used to herd cattle under the direction of their owner, or dogs that are being used for hunting or predator control under the direction of their owner.
This article is intended to be helpful and informative. But even common legal matters can become complex and stressful. A qualified pet lawyer can address your particular legal needs, explain the law, and represent you in court. Take the first step now and contact a local pet attorney to discuss your specific legal situation.