Here is what I mean. My character- Violet- started the play with a lofty attitude, but now her pride has been broken by her mother's death and betrayal of a friend. The scene begins with her reuniting with her brother- and she needs to tell him what has happened. I want the brother to become angry at their mother's killer. And I want Violet to come across more polite and empathetic. But how do I do that? At the end of the scene, both characters should be determined to fight the one who killed their mother.
Embrace the conflict. Just like a screenplay, a scene can have an ACT I, ACT II, ACT III.
And people get absorbed in conflict, there should be some conflict in every scene, and your scene is built for conflict.
The Acts within a scene are obviously much shortened, perhaps to a single scene, but they still exist. For yours (call the brother Bob):
Scene ACT I, first half (12.5% of scene) Bob's "normal world". A scene of her brother doing something normal when she arrives; mowing the lawn, practicing fly fishing, whatever. Something that tells us WHAT KIND OF GUY BOB IS. Since he gets angry, you might want to foreshadow this in the scene: He's teaching his kid to fight, and they seem to be having fun at it. Or he is watching a martial arts contest, or shooting competition.
Scene Act I, midpoint: The inciting incident: Violet arrives.
Scene Act I, second half: Inciting incident escalates; Violet is crying; Bob sees it, asks what is wrong. "Mom was murdered. She's dead." This escalates, Bob can't believe it. Asks questions. Scene Act I ends with "Bob leaving his normal world," he cannot continue his routine, he is focused on his mother's murder.
We are 25% of the way through the scene. On to:
Scene Act II, first half: Bob's reactive phase. Bob expresses grief through livid anger. He wants to bring the killer to justice, punishment, death, what have you. He's a fighter, he wants vengeance. Torture even. Violet provides the resistance; reasons why Bob's impulsive reactions won't work, or will get him killed.
We have reached the halfway point in the scene (50%):
Scene Act II, second half: Bob's proactive phase. He still wants to bring the killer to justice; but having been thwarted (sensibly) by Violet, his brain starts working again. More planning on how to bring the killer to justice. Still a few objections by Violet (setbacks for Bob), but he is adapting to the problem. At the end of this Act, they discover the idea or clue to what they need to do.
We have reached the 75% point in the scene, and ACT III (climax and resolution):
Scene Act III, first half, climax (15% of total scene): Bob and Alice use the clue from Act II to develop their plan of what to do next, one they agree upon.
Scene Act III, second half, emotional resolution (10% of total scene): They know what to do, Bob's initial anger reaction has subsided with the climax; I'd reassert his grief at the loss of his mother, as the pair set off on their new mission.
Even if your whole scene is just two pages long, stick to the three act structure within that limit. Bob has a character arc to go through here. Basically all of this scene is part of an ACT I: We watch Bob go from his normal world to leaving it on a mission. If the scene is pretty short, you have to convey this stuff visually and with just a few lines of dialogue.
For example, in a two page scene, 1/8 of it is a quarter of a page. Not a lot of wiggle room!
But anger and grief are easy to show visually (music will aid it); Bob can throw or slam something, his expression can convey grief and rage. Violet's reaction to this can likewise convey her restraint and determination to get him over this phase.
You have to trust your actors; you give them the clues to what the character is feeling; don't be too specific in describing expressions; instead pick your words describing the emotion precisely and don't be too vague ("angry" is fairly vague; there are many types of anger; from cold to hot, quiet to loud, calm to violent). Actors will find the expressions to convey the precise words.