In the head would be most effective and painless but messy. A vet should look at the dog first before it's put down and if it still needs to be, most vets will euthanize a dog for free since it's usually a traumatic event for the owners.
Where Is The Proper Place To Shoot A Dog To Give It A Quick Death, Due To An Accident?
Well you can shoot your dog near grave yard and then instantly burry this dog in a grave. This is the proper way to get rid of any mad dog.