Are the body and clothes of one who fondles stray dogs he feeds considered impure, and how can he purify himself to resume his prayers?

Feeding stray dogs is permissible, and the one who does so will be rewarded for it, based on the hadith: "In every wet liver is a reward." However, playing with them and petting them is not advisable due to their dirtiness, bad odor, and the impurity of their saliva, in addition to the fact that angels do not enter a house that has a dog. Dog hair is pure, but its saliva is impure and requires purification if it gets on clothes.