die Dreistigkeit
audacity
The quality of being disrespectful, insolent, or cheeky while deliberately disregarding other people's feelings or rights.
Seine Dreistigkeit kannte keine Grenzen.
His audacity knew no bounds.

audacity / insolence
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audacity
The quality of being disrespectful, insolent, or cheeky while deliberately disregarding other people's feelings or rights.
Seine Dreistigkeit kannte keine Grenzen.
His audacity knew no bounds.

The noun 'die Dreistigkeit' is derived from the adjective 'dreist', which means 'insolent' or 'cheeky'. The ending '-keit' forms nouns from adjectives that describe a quality.
Er ist bekannt für seine dreiste Art.
He is known for his cheeky manner.
'Dreistigkeit' can be uncountable and describe the general quality. But it can also be countable and refer to a single, concrete insolent action; in that case, a plural ('Dreistigkeiten') is also possible.
Seine Dreistigkeit ist unglaublich. (Eigenschaft)
His audacity is unbelievable. (quality)
When describing what the audacity consists of, an infinitive construction with 'zu' is often used. Typical phrases are 'die Dreistigkeit besitzen/haben, etwas zu tun' (to have the audacity to do something).
Er hatte die Dreistigkeit, nach all den Lügen wieder anzurufen.
He had the audacity to call again after all those lies.
When a single action is meant, the verb 'sich erlauben' (to allow oneself / to dare to do something) is often used. The expression is then 'sich eine Dreistigkeit erlauben' (to commit an act of insolence).
Wie kannst du es wagen, dir eine solche Dreistigkeit zu erlauben?
How dare you commit such an act of insolence?
impudence / cheek
'Frechheit' and 'Dreistigkeit' are very close in meaning. Both can mean disrespectful behavior or a single brazen act. 'Dreistigkeit' often puts even stronger emphasis on someone doing something deliberately and without shame, even though it harms or annoys others. 'Frechheit' is very common in everyday language and can also be used for an impolite remark. Learners may confuse the words because both fit similar situations. If you especially mean the bold, shameless nature of an action, 'Dreistigkeit' is more precise.

Hannes hatte die Dreistigkeit, sich an der Kasse einfach vorzudrängen.
Hannes had the audacity to just cut in line at the checkout.
Im Krankenhaus war es eine Frechheit, dass der Besucher das Personal laut beschimpfte.
At the hospital, it was outrageous that the visitor shouted insults at the staff.