rage
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in a rage
In a state of extreme, uncontrolled anger or fury. The phrase describes that someone becomes or is very angry.
Die ungerechte Behandlung brachte ihn völlig in Rage.
The unfair treatment sent him completely into a rage.
The noun 'Rage' is used almost exclusively in the fixed expression 'in Rage' (in a rage). Using it by itself with an article ('die Rage') is very rare and sounds unnatural.
Seine unfairen Vorwürfe brachten sie sofort in Rage.
Her unfair accusations immediately sent her into a rage.
The phrase 'in Rage' is combined with three typical verbs: 'jemanden in Rage bringen' (to send someone into a rage), 'in Rage geraten' (to fly into a rage; passive/sudden), and 'in Rage sein' (to be in a rage; state).
Als er die Lüge erfuhr, geriet er in Rage.
When he found out about the lie, he flew into a rage.
'In Rage sein' (to be in a rage) describes very strong, often uncontrolled anger. It is emotionally much more intense than simply 'wütend sein' (to be angry) or 'sich ärgern' (to be annoyed).
Er schrie und schlug mit der Faust auf den Tisch, so sehr war er in Rage.
He shouted and slammed his fist on the table, he was so much in a rage.
annoyance / trouble
'der Ärger' and 'die Rage' both have to do with anger. But 'Ärger' is much more general: you can be annoyed about something without completely losing control. In addition, 'Ärger' can also mean problems or difficulties. 'Rage' is used almost only in the phrase 'in Rage' and means very strong, uncontrolled anger. Learners may confuse the words because both describe negative feelings. In practical terms: for normal annoyance or problems, 'Ärger' fits; for extreme anger, 'in Rage' fits.
Nach der Beleidigung geriet der Kunde völlig in Rage.
After the insult, the customer flew completely into a rage.
Wegen der falschen Rechnung haben wir jetzt viel Ärger.
Because of the incorrect bill, we now have a lot of trouble.