nuisance / annoyance
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nuisance
A thing, person, or circumstance that causes annoyance, displeasure, or irritation.
Der ständige Baulärm ist ein großes Ärgernis für die Anwohner.
The constant construction noise is a major nuisance for the residents.
'Das Ärgernis' (the nuisance/annoyance) belongs to the word family of the verb 'ärgern' (to annoy). The suffix '-nis' forms a noun here that describes the thing or circumstance that causes annoyance.
Die Verspätung des Zuges war ein Ärgernis für alle Pendler.
The train delay was a nuisance for all commuters.
The word 'Ärgernis' (nuisance/annoyance) is used mainly in written language or in a more formal style. It sounds more elevated than everyday expressions for annoyance.
Der Müll auf der Straße stellt ein hygienisches Ärgernis dar.
The garbage on the street constitutes a hygienic nuisance.
If you want to express for whom something is a nuisance, you use the preposition 'für' (for) followed by the accusative.
Die laute Musik in der Nacht war ein Ärgernis für die Nachbarn.
The loud music at night was a nuisance for the neighbors.
In the meaning of 'Anstoß erregen' (to cause offense), the word is almost always used in the fixed expression 'öffentliches Ärgernis erregen' (to cause a public nuisance). This formulation is typical of legal or official contexts.
Durch sein Verhalten auf dem Marktplatz erregte der Mann öffentliches Ärgernis.
Through his behavior in the marketplace, the man caused a public nuisance.
annoyance / trouble
'der Ärger' and 'das Ärgernis' are closely related, but they do not refer to the same thing. 'Ärger' is the feeling someone has: you are angry, dissatisfied, or annoyed. 'Ärgernis' is usually the thing, person, or situation that causes this feeling. So you can say: the 'Ärgernis' causes 'Ärger'. The words are easy to confuse because both have to do with irritation or annoyance. If you are talking about the feeling, use 'Ärger'; if you are talking about the reason for it, 'Ärgernis' fits better.
Der ständige Lärm im Treppenhaus ist für die Nachbarn ein großes Ärgernis.
The constant noise in the stairwell is a major nuisance for the neighbors.
Nach dem verpassten Termin konnte Paul seinen Ärger kaum verbergen.
After missing the appointment, Paul could hardly hide his annoyance.