
bügeln
to iron
To smooth clothing or other textiles with a hot iron in order to remove wrinkles.
Ich muss noch meine Hemden für die nächste Woche bügeln.
I still have to iron my shirts for next week.
to iron / to smooth over
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to iron
To smooth clothing or other textiles with a hot iron in order to remove wrinkles.
Ich muss noch meine Hemden für die nächste Woche bügeln.
I still have to iron my shirts for next week.

to smooth something over
To fix a mistake, problem, or mishap and eliminate the negative consequences.
Er hat einen großen Fehler gemacht, und nun muss seine Assistentin das wieder für ihn ausbügeln.
He made a big mistake, and now his assistant has to smooth it over for him.
The verb 'bügeln' (to iron) is closely related to the nouns 'das Bügeleisen' (the iron, the device for smoothing) and 'der Bügel' (e.g. a clothes hanger). The act of ironing creates smooth surfaces and clean edges.
Kannst du mir zeigen, wie man das Bügeleisen richtig benutzt?
Can you show me how to use the iron properly?
In its literal meaning, 'bügeln' (to iron) is often combined with the textiles that are being smoothed. Typical examples are 'Wäsche bügeln' (to iron laundry), 'ein Hemd bügeln' (to iron a shirt), or 'eine Bluse bügeln' (to iron a blouse).
Am Sonntagabend bügle ich immer meine ganze Wäsche für die Woche.
On Sunday evenings, I always iron all my laundry for the week.
In the figurative meaning 'to correct a mistake,' 'ausbügeln' (to smooth over/fix) is a separable verb. The prefix 'aus-' stands at the end in a main clause.
Mein Kollege hat den Kunden verärgert, und ich musste seinen Fehler wieder ausbügeln.
My colleague upset the customer, and I had to smooth over his mistake again.
In the figurative meaning, 'ausbügeln' (to smooth over/fix) refers to abstract problems. Common combinations are 'einen Fehler ausbügeln' (to fix a mistake), 'ein Missverständnis ausbügeln' (to clear up a misunderstanding), or 'ein Problem ausbügeln' (to fix a problem).
Sie versuchte, das peinliche Missverständnis schnell wieder auszubügeln.
She tried to quickly smooth over the embarrassing misunderstanding.
to smooth
Bügeln means smoothing textiles with a hot iron. Glätten describes removing wrinkles more generally and can also be done by hand or by another method. If the iron is what matters, bügeln is the right choice; if the focus is only on the result, glätten is appropriate.

Sie bügelt das zerknitterte Hemd mit einem heißen Bügeleisen.
She is ironing the wrinkled shirt with a hot iron.
Sie glättet das leicht zerknitterte Hemd vorsichtig mit den Händen.
She is carefully smoothing the slightly wrinkled shirt with her hands.