
billigen
to approve
To officially approve, endorse, or agree to something. Often used in formal or official contexts.
Der Stadtrat hat den Bau des neuen Kindergartens gebilligt.
The city council approved the construction of the new kindergarten.
to approve / condone
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to approve
To officially approve, endorse, or agree to something. Often used in formal or official contexts.
Der Stadtrat hat den Bau des neuen Kindergartens gebilligt.
The city council approved the construction of the new kindergarten.

to condone
To tacitly tolerate or accept negative or questionable behavior, even though you do not think it is good.
Wer Gewalt nicht aktiv verurteilt, scheint sie stillschweigend zu billigen.
Anyone who does not actively condemn violence seems to tacitly condone it.
The verb 'billigen' (to approve / condone) is related to the adjective 'billig' (cheap). 'Billig' originally meant 'fair' or 'appropriate'. From this older meaning, 'billigen' developed in the sense of 'to recognize something as fair' or 'to approve'.
Die Aktionäre billigten den Vorschlag des Vorstands.
The shareholders approved the board's proposal.
In the meaning of 'genehmigen' (to approve), 'billigen' belongs to formal official or business language. You often find it in minutes, contracts, or official announcements.
Das Parlament muss das neue Gesetz noch billigen.
Parliament still has to approve the new law.
Formal plans or decisions are typically approved. Common objects are: 'einen Antrag billigen' (to approve an application/motion), 'ein Budget billigen' (to approve a budget), 'einen Plan billigen' (to approve a plan), or 'ein Vorgehen billigen' (to approve a course of action).
Die Geschäftsführung hat die Investition gebilligt.
Management approved the investment.
In this meaning, 'billigen' (to condone) has a negative connotation and is often combined with the adverb 'stillschweigend' (tacitly). It describes not actively preventing or criticizing problematic behavior and thereby tolerating it.
Er hat die unfairen Kommentare seines Kollegen stillschweigend gebilligt.
He tacitly condoned his colleague's unfair comments.
to accept
Both words can refer to accepting a plan or result. „Akzeptieren“ can mean merely putting up with something without thinking it is good. „Billigen“, on the other hand, includes a positive evaluation or explicit approval and is appropriate when you endorse the matter.

Ich billige den Plan, weil er für alle fair ist.
I approve of the plan because it is fair to everyone.
Ich akzeptiere den Plan, obwohl ich ihn nicht gut finde.
I accept the plan, even though I do not like it.