abschlagen
to refuse
To reject or turn down a request, offer, or wish.
Er konnte ihr diese Bitte unmöglich abschlagen.
He simply could not refuse her this request.

to refuse / chop off
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to refuse
To reject or turn down a request, offer, or wish.
Er konnte ihr diese Bitte unmöglich abschlagen.
He simply could not refuse her this request.

'Abschlagen' (to refuse/chop off) is a separable verb. In main clauses, the prefix 'ab' comes at the end of the sentence. In subordinate clauses or in the infinitive with 'zu', the verb stays together.
Der Ritter schlug dem Drachen den Kopf ab.
The knight chopped off the dragon's head.
The verb consists of 'ab' (away, down/off) and 'schlagen' (to hit/strike). The literal meaning 'to strike something away' or 'to knock something down/off' is the basis for almost all uses, including the figurative meaning 'ablehnen' (to refuse/reject).
Bevor die Wand neu gestrichen wird, müssen wir die alte Farbe abschlagen.
Before the wall is repainted, we have to chip off the old paint.
When refusing a request, the person whose wish is being refused is often in the dative. The pattern is: jemandem (dative) etwas (accusative) abschlagen (to refuse someone something).
Die Bank hat ihm den Kredit leider abgeschlagen.
Unfortunately, the bank refused him the loan.
In this meaning, 'abschlagen' (to tee off) is a technical term from golf. It refers to the first stroke from a special starting point (the teeing area) and is used almost only in this context.
Die Spieler warten darauf, am nächsten Loch abschlagen zu können.
The players are waiting to be able to tee off at the next hole.
The meaning 'abwehren' (to repel/fend off) is typically used in a military or combat context. Common objects are 'der Angriff' (the attack), 'der Feind' (the enemy), or 'die Attacke' (the attack).
Die Armee konnte den nächtlichen Angriff erfolgreich abschlagen.
The army was able to successfully repel the nighttime attack.
to accept / assume
'annehmen' and 'abschlagen' are opposites here. If you 'annimmt' an offer, request, or wish, you say yes to it. If you 'abschlägt' something, you turn it down. 'annehmen' sounds neutral and is very common. 'abschlagen' sounds somewhat more elevated or firm than simply 'ablehnen' (to reject), but it fits well with offers or requests. So the difference is clear: 'annehmen' means to agree or accept; 'abschlagen' means to reject or refuse.

Der Kunde schlägt das Angebot ab, weil es ihm zu teuer ist.
The customer turns down the offer because it is too expensive for him.
Nach kurzer Überlegung nahm sie die Einladung zum Gespräch an.
After thinking it over briefly, she accepted the invitation to the meeting.