
angenommen
assuming
Used as a particle to introduce a hypothetical situation or assumption; in the sense of 'supposing that...'.
Angenommen, wir verpassen den Zug – was machen wir dann?
Assuming we miss the train — what do we do then?
assuming / accepted
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assuming
Used as a particle to introduce a hypothetical situation or assumption; in the sense of 'supposing that...'.
Angenommen, wir verpassen den Zug – was machen wir dann?
Assuming we miss the train — what do we do then?

accepted
As an adjective, describes something that has been accepted, approved, or received.
Der angenommene Gesetzesentwurf wird nun im Parlament debattiert.
The accepted draft law will now be debated in parliament.
'Angenommen' is the past participle of the verb 'annehmen'. This verb has two main meanings: 'to suppose/think hypothetically' and 'to accept something'. The two uses of 'angenommen' follow directly from these meanings.
Ich nehme an, du kommst morgen zur Party.
I assume you're coming to the party tomorrow.
You can recognize the meaning from the sentence position. If 'Angenommen' (assuming) is at the beginning of the sentence, followed by a comma, it introduces a hypothesis. If it appears before a noun, it functions as an adjective and describes that noun's accepted state.
Angenommen, das Wetter ist schlecht, dann grillen wir drinnen.
Assuming the weather is bad, we'll grill indoors.
When 'Angenommen' (assuming) introduces a hypothesis, it stands in position 1. After that comes a comma and a normal main clause, with the verb in position 2.
Angenommen, du gewinnst im Lotto, was würdest du kaufen?
Assuming you win the lottery, what would you buy?
In the meaning of 'accepted', the word is often used in formal or administrative contexts. Typical combinations are 'ein angenommener Antrag' (an accepted application/motion), 'ein angenommenes Gesetz' (an adopted law), or 'ein angenommenes Angebot' (an accepted offer).
Nach der Abstimmung galt der Antrag als angenommen.
After the vote, the motion was considered accepted.
approved
Genehmigt means that a responsible authority has officially approved an application or project. Angenommen is broader and, with an application, can simply mean that it was received or accepted. Acceptance alone therefore does not always mean that approval has already been granted.

Der Antrag wurde am Schalter angenommen, aber noch nicht geprüft.
The application was accepted at the counter, but it had not yet been reviewed.
Nach der inhaltlichen Prüfung wurde der Antrag genehmigt.
After the application was reviewed in detail, it was approved.