When something happens rightfully, it happens the way it should — because the person has a real claim, because it is fair, or because the law says so. You reach for it when you want to say “this is not just what they fancy; they deserve it,” or “this is the proper, honest way.” A child who rightfully inherits a house is the one the will (or the law) names, not the cousin who simply moved in. A worker who is rightfully angry was treated badly, and the anger makes sense. Though actually, people stretch it: “She rightfully belongs on that team” can just mean “anyone with sense would put her there,” even when no court is involved. It is a little more formal than “fairly.” It is not the same as “rightly,” which more often means “correctly” — you rightly guessed the answer; you rightfully kept the money that was yours.