
die Begegnung
encounter
A usually accidental meeting with one or more people.
Die Begegnung mit meinem alten Schulfreund auf der Straße hat mich sehr gefreut.
The encounter with my old school friend on the street made me very happy.
encounter / match
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encounter
A usually accidental meeting with one or more people.
Die Begegnung mit meinem alten Schulfreund auf der Straße hat mich sehr gefreut.
The encounter with my old school friend on the street made me very happy.

match
A sporting contest between two teams or individuals.
Die Begegnung zwischen Bayern München und Borussia Dortmund endete unentschieden.
The match between Bayern Munich and Borussia Dortmund ended in a draw.
'Die Begegnung' (encounter/match) comes from the verb 'begegnen' (to encounter/meet), which always requires the dative. This grammar often carries over to the noun: people speak of 'Begegnung mit jemandem' (encounter with someone, dative).
Ich begegne meinem Nachbarn jeden Morgen.
I meet my neighbor every morning.
Whether it means a chance meeting or a sports contest is easy to recognize from context. If the sentence is about people in everyday life, it is an encounter. If it is about sports, teams, or a competition, it is a match.
Es war eine rein zufällige Begegnung am Bahnhof.
It was a purely chance encounter at the train station.
When it is about a meeting, 'Begegnung' (encounter) is often described with adjectives like 'zufällig' (chance/accidental), 'unerwartet' (unexpected), 'kurz' (brief), 'flüchtig' (fleeting), or 'unvergesslich' (unforgettable).
Es war nur eine flüchtige Begegnung, aber ich erinnere mich an ihr Lächeln.
It was only a fleeting encounter, but I remember her smile.
In a sports context, the structure 'die Begegnung zwischen [Team A] und [Team B]' (the match between [Team A] and [Team B]) is almost always used to name the two opponents.
Millionen von Zuschauern verfolgten die Begegnung zwischen den beiden Rivalen.
Millions of viewers watched the match between the two rivals.
game / play
Spiel and Begegnung can refer to the same sporting contest. Spiel is the neutral term and can place particular emphasis on the actual course of the game. Begegnung often sounds journalistic and focuses more strongly on the scheduled contest between the opponents.

Die Begegnung zwischen Köln und Mainz findet am Samstag statt.
The match between Cologne and Mainz will take place on Saturday.
Das Spiel zwischen Köln und Mainz wurde in der zweiten Hälfte spannend.
The game between Cologne and Mainz became exciting in the second half.