Anger, incense, enrage, infuriate and madden all these verbs carry in common with anger, their general term, the denotation to make angry or to rouse to anger.
- laugh then at any, but at fools or foes; these you but anger, and you mend not those
—Pope - angered by his son’s repeated disobedience
Incense implies hotness of anger, especially as provoked by something excessively irritating and offensive
- Mr. Critchlow, aged and unaccustomed to interference, had to render accounts of his trusteeship to this young man, and was incensed
—Bennett - magistrates and populace were incensed at a refusal of customary marks of courtesy and respect for the laws
—Inge
Enrage suggests a violent display of wrath or fury.
- I pray you, speak not . . . question enrages him
—Shak.
Infuriate may imply a sense of being outraged or sometimes no more than of being thoroughly irritated or exasperated.
- how it infuriates a bigot, when he is forced to drag into the light his dark convictions!
— L. P. Smith - his colleagues and his subordinates had been alternately delighted and infuriated by his assumed reluctance to deal with any practical question
—Sackville-West
Madden is often not distinguishable from infuriate.
- can it be fancied that Deity ever vindictively made in his image a mannikin merely to madden it?
—Poe
Like the former it may imply merely excessive annoyance or vexation.
- maddening delays