
entgegen
toward
In the direction of someone or something; moving toward someone or something.
Sie ging ihrem Bruder entgegen.
She walked toward her brother.
toward / contrary to
entgegen + dative
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toward
In the direction of someone or something; moving toward someone or something.
Sie ging ihrem Bruder entgegen.
She walked toward her brother.

contrary to
In opposition to something or against an expectation, rule, or instruction.
Entgegen den Anweisungen öffnete er die Tür.
Contrary to the instructions, he opened the door.
'Entgegen' usually takes the dative. The person or thing one is moving toward, or whose instruction one is contradicting, therefore appears in the dative.
Die Wanderer liefen dem Wind entgegen.
The hikers walked toward the wind.
Entgegen meinem Rat kaufte sie das teure Auto.
Contrary to my advice, she bought the expensive car.
In its directional meaning, 'entgegen' usually appears as a verb particle, for example in 'entgegengehen' (to walk toward), 'entgegenkommen' (to come toward), or 'entgegenfahren' (to drive toward). The person or thing appears in the dative; as a preposition, 'entgegen' with the dative is mainly used in the meaning 'contrary to'.
Am Bahnhof kam mir meine Freundin entgegen.
My friend came toward me at the train station.
In the meaning 'contrary to', 'entgegen' often combines with 'Anweisungen' (instructions), 'Regeln' (rules), 'Vorschriften' (regulations), or 'Erwartungen' (expectations). A particularly common combination is 'entgegen allen Erwartungen' (contrary to all expectations).
Entgegen allen Erwartungen gewann die junge Mannschaft das Finale.
Contrary to all expectations, the young team won the final.