All perfect praise be to Allah, The Lord of The Worlds, and may His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
Islam is the religion of mercy, and Almighty Allah has sent Prophet Mohammad(PBUH) as a mercy for all creatures where He said{What means}, "We sent thee not, but as a mercy for all creatures"{Al-Anbia`, 107}. Therefore, this mercy isn`t only for man and animals; rather, it is for all creatures.
Sharia scripts teach that kindness and forbearance should be observed in everything, even with animals, since the Messenger of Allah was reported to have said, "A woman got into Hell-Fire because of a cat whom she had tied, and thus it could not eat, and she did not let it free so that it could devour the vermin of the earth, until it died."{Related by Bukhari}. He(PBUH) also said, "Allah's Messenger (PBUH) said, "While a man was walking he felt thirsty and went down a well and drank water from it. On coming out of it, he saw a dog panting and eating mud because of excessive thirst. So he (went down the well), filled his shoe with water, caught hold of it with his teeth and climbed up and watered the dog. Allah thanked him for his (good) deed and admitted him to heaven." {Related by Ibn Hibban}.
Among the key objectives of Sharia and on top of its priorities is preserving man`s life, wealth and honor, as a means of honoring him/her. On this basis, Sharia has allowed killing some predatory animals that pose a threat to man`s life, for the Prophet(PBUH) said, "Five are the beasts in killing which or their being killed, there is no sin: crow, kite, scorpion, rat and voracious dog."{Related by Bukhari}.
Al-Imam An-Nawawi said, "According to the majority of the Muslim scholars, rabid dog refers to every ferocious animal, such as the jaguar, the tiger, the wolf and the like."{Shareh An-Nawawi Ala Muslim 8/114-115}.
In conclusion, it is permissible to ward off the harm of rabid dogs and similar animals as a means of honoring man whose life and wealth are preserved by Sharia. But, if their harm couldn't be prevented save by putting them down, then doing so is permissible. However, man should observe kindness when doing that because the Prophet(PBUH) said, "Verily Allah has prescribed ihsan (proficiency, perfection) in all things. So if you kill then kill well; and if you slaughter, then slaughter well. Let each one of you sharpen his blade and let him spare suffering to the animal he slaughters.”{Related by Muslim}. And Allah knows best.