to annoy / to get annoyed
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to get annoyed
To be angry or annoyed about a person, thing, or situation.
Ich ärgere mich jeden Morgen über den dichten Verkehr auf dem Weg zur Arbeit.
I get annoyed every morning about the heavy traffic on the way to work.
Pay attention to the difference: 'sich ärgern' (to get annoyed) is reflexive and means that you yourself are angry. 'Jemanden ärgern' (to annoy someone) is transitive and means that you deliberately make another person angry.
Ich ärgere mich über das schlechte Wetter.
I get annoyed about the bad weather.
The verb 'ärgern' (to annoy / to get annoyed) is closely related to the noun 'der Ärger' (anger, trouble, annoyance). When you get annoyed, you have Ärger. When you annoy someone, you cause them Ärger.
Bitte mach mir keinen Ärger!
Please don't cause me any trouble!
When saying what you are angry about, you always use the reflexive form 'sich ärgern' (to get annoyed) with the preposition 'über' followed by the accusative.
Sie ärgert sich über ihren lauten Nachbarn.
She gets annoyed about her noisy neighbor.
When you deliberately provoke someone, you use 'ärgern' (to annoy) with a direct object in the accusative. This object is the person who is being annoyed.
Hör auf, deine kleine Schwester zu ärgern!
Stop annoying your little sister!
to disturb / bother
'stören' and 'ärgern' can both have to do with an unpleasant situation. 'stören' more often means that something interrupts peace and quiet, concentration, or an activity, or is felt to be annoying. 'ärgern' emphasizes the feeling: you are angry or annoyed. In addition, 'jemanden ärgern' can mean deliberately provoking or teasing someone. The words are easy to confuse because a disturbance often also causes annoyance. But not everything that 'stört' makes you truly angry, and annoyance can also arise without a direct interruption.
Sofia ärgert sich über den Fehler in der Rechnung.
Sofia is annoyed about the mistake in the invoice.
Bitte stör mich nicht, ich muss diese E-Mail fertig schreiben.
Please don't disturb me; I need to finish writing this email.