Which figure of speech is used in this text?
There was a clear consensus in the Vance family that they should put their dog to sleep, rather than let him continue to suffer.

 - euphemism
 - paradox
The text uses a euphemism, a polite or indirect expression that is used to de-emphasize an unpleasant topic.
Put their dog to sleep is a more indirect way of saying have the veterinarian kill their dog.