Whether your dog is branded as a dangerous dog and as such is euthanized varies from state to state and even city to city.
In the city I live in, in northern MN, a dog is automatically labeled as dangerous if they attack a human...however, nothing really happens if they hurt of kill another dog. There might be a fine, but the dog isn't euthanized unless they're a repeat offender (meaning it's not the first dog it killed).
The best bet to find out what the rules and statutes are in your area is to call your city clerk's office and ask them for the statutes on dogs at large...this will tell you what could happen with your dog.
I hope this helps.