"Angry" is a blur. Underneath it sit fifteen sharper words, and they are not interchangeable. Some are mild, some are explosive. Some carry a sense of unfairness, some a childish sulk, some a quiet, dangerous restraint. Here is the anger cluster from 1983, laid out from mild to furious.
Mildly annoyed by something trivial.
low-grade irritation
Fretful and quick to complain; ill-tempered over small things.
querulous, hard to please
Childishly sulky and bad-tempered, like someone who did not get their way.
a thwarted child's anger
Sullen and simmering; unhappy with how one has been treated.
quiet dissatisfaction
Anger at something perceived as unfair or unjust.
principled anger, the moral one
Hot, blazing anger; loud and direct.
heat, no subtext
Offense taken at a slight, real or imagined (you "take umbrage").
wounded dignity
Habitually bad-tempered and difficult by disposition.
a standing temperament, not a moment
Aggressively defiant and belligerent; quick to argue or fight.
looking for a fight
Anger barely suppressed; burning with no open flame.
a slow burn
Burning with indignation; furious at something offensive.
heat plus moral outrage
Extreme anger expressed with violent intensity.
white-hot, outward
Boiling with anger that is being held in.
intense but suppressed
Intense, often righteous anger.
a biblical register
So enraged it is framed as physical, as if about to have a stroke.
the ceiling
All three are "very angry," but they differ on the nuance axis. Indignant is anger at unfairness, more principle than heat. Irate is pure hot fury, loud and personal, with no moral subtext. Incensed is stronger than both and morally charged: indignation that has caught fire. By intensity: indignant, then irate, then incensed.
Both are low-grade irritation. Peevish is fretful and complaining, the mood of someone hard to please. Petulant is sulky and childish, the anger of someone who did not get their way. Peevish nags; petulant pouts.
Same extreme intensity, opposite expression. Seething is anger held in, quiet and dangerous. Apoplectic is anger that has burst out completely. One is a closed valve, the other a blown one.
Incensed is the more intense word, and it carries moral outrage: it literally means burning with indignation. Irate is hot, blazing anger without that charge. On the 1983 intensity scale irate sits at 6 and incensed at 8, so incensed is both stronger and more morally loaded.
No. Indignant specifically means anger at something unfair or unjust. You can be furious without being indignant, and indignant without being especially intense. The unfairness is the defining feature, not the heat.
Seething, smouldering, and disgruntled all describe suppressed anger. Seething is the most intense, smouldering is a slow burn, and disgruntled is sullen and low-grade.
Apoplectic sits at the top of this cluster, describing rage so extreme it is framed as physical. Wrathful and seething sit just below it.
1983 maps the anger cluster, and hundreds of others, as a field you can tap through, with recorded audio for every word and quizzes that test whether you can place a word among its neighbors.