Your dog' s tail is wagging because he is exited or likes
something you did.
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The possessive form is: The wet dog's tail is wagging.
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It can, but this is more likely when the dog is sitting or lying
down. When a dog is standing and wagging its tail, it's the
momentum of the tail wagging that spreads to the rest of the dog's
body and causes the rest of the dog to wag too.