When one party is a child, we as the adult bear full responsibility for calling off the battle by flatly refusing to fight. Remember, no fight can continue unless both sides are willing to participate. Initially, it takes patience and self-control to resist matching anger with anger in return. But once mastered, the results of your new approach will be much more rewarding. To avoid reacting on impulse requires more effort in that we have to tune in more to the message behind the words than to the words themselves.
Anger is often a disguise for pain.
By refusing to be detoured by the anger, and by responding directly to the pain, you show the children that you aren't fooled by their bluster because you're hip to what's really going on inside them. If you speak to the hurt and ignore the anger, the anger often goes away on its own. I obviously care more about you than she does. It's hard not knowing where you belong or if anyone really cares, but you know I took you in because I love you, not because I had to. There are no appropriate or inappropriate emotions, just appropriate and inappropriate ways of expressing or dealing with them.