
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.

act of insolence
A single, concrete action that is perceived as insolent or cheeky.
Sich in der Schlange einfach vorzudrängeln, ist eine unglaubliche Dreistigkeit.
Simply cutting in line is an unbelievable act of insolence.
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 refer to disrespectful, insolent behavior. Frechheit focuses more on the disrespectful effect of an action; Dreistigkeit emphasizes more strongly that someone acts very boldly and deliberately, without consideration for others.

Mit großer Dreistigkeit bleibt er trotz aller Proteste vor den Wartenden stehen.
With great audacity, he continues standing at the front of the queue despite all the protests.
Dass er sich ohne Entschuldigung vor die Wartenden stellt, ist eine Frechheit.
It's outrageous that he cuts in front of the people waiting without apologizing.