hang-up / lapse / trailer
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lapse
Informal term for a brief, unexpected lapse in memory or concentration, especially when speaking or thinking. You suddenly no longer know how to continue.
Mitten im Vortrag hatte ich einen totalen Hänger und wusste nicht mehr, was ich sagen wollte.
In the middle of the presentation, I had a total lapse and no longer knew what I wanted to say.
The word 'Hänger' comes from the verb 'hängen' (to hang). The basic idea is that something is stuck or not moving forward. This fits both a memory that 'hängt' (gets stuck) and a person who is listlessly 'durchhängt' (slumps).
Ich glaube, die Software hat sich aufgehängt.
I think the software has frozen.
All meanings of 'Hänger' are colloquial. In formal or written contexts, you would usually use more precise expressions.
Sorry, ich hatte gerade einen kleinen Hänger.
Sorry, I just had a little lapse.
For a memory lapse and a motivational slump, the fixed expression 'einen Hänger haben' (to have a lapse/slump) is almost always used. The verb 'haben' (to have) is the key partner here.
Ich lerne den Text auswendig, damit ich in der Prüfung keinen Hänger habe.
I'm memorizing the text so that I don't have a lapse in the exam.
You can recognize the meaning 'Anhänger' (trailer) from the context. If it is about cars, transport, or pulling loads, the vehicle is meant.
Kannst du mir deinen Hänger leihen? Ich muss zum Baumarkt.
Can you lend me your trailer? I need to go to the hardware store.
supporter / trailer / pendant
'der Anhänger' and 'der Hänger' can both mean a vehicle without its own engine that is pulled by a car or truck. In this meaning, they are practically synonyms. 'Anhänger' is the normal and neutral form, especially in official or clear situations. 'Hänger' is more colloquial and fits better in everyday speech. They can also be confused because 'Hänger' has other colloquial meanings, for example a brief mental lapse. If you want to make it clear that you mean the vehicle, 'Anhänger' is safer.
Den Hänger holen wir morgen früh in der Werkstatt ab.
We'll pick up the trailer from the garage tomorrow morning.
Wegen des Umzugs mietet Paul einen Anhänger für das Auto.
Because of the move, Paul rents a trailer for the car.