Is it permissible to be courteous to an ill-mannered person in daily interactions by speaking with him—despite the anger and rage in one's heart and praying for Allah to avenge oneself against him—out of consideration for the general welfare of the family, and is this considered hypocrisy?

It is legislated for a Muslim to deal with people kindly and gently, even if there is anger in their heart towards them due to their injustice, and this is not considered hypocrisy. Al-Bukhari reported from Abu Darda’ his saying: "Indeed, we smile in the faces of some people, and yet our hearts curse them." And from Aisha, that the Prophet, peace and blessings be upon him, said to a man: "What a terrible son of his tribe," then he softened his speech towards him, justifying this by saying that "the worst of people in station before Allah is he whom people abandon to avoid his indecency." And it is more befitting for a Muslim to pardon and forgive those who wrong them among relatives and neighbors, and to repel their wrongdoing with good deeds.