der Fang
catch
The total number of animals, especially fish or game, that were caught during a hunt or fishing trip.
Der Fischer verkaufte seinen gesamten Fang direkt am Hafen.
The fisherman sold his entire catch directly at the harbor.

catch / fang
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catch
The total number of animals, especially fish or game, that were caught during a hunt or fishing trip.
Der Fischer verkaufte seinen gesamten Fang direkt am Hafen.
The fisherman sold his entire catch directly at the harbor.

The noun 'der Fang' (catch) is derived directly from the verb 'fangen' (to catch). This connection helps you understand most meanings: it can refer to the animals caught, the act of catching, or a 'caught' (won over) valuable person.
Der Angler war stolz auf seinen Fang.
The angler was proud of his catch.
In the figurative meaning, the fixed expression 'einen guten Fang machen' (to make a good catch) is almost always used. It describes gaining a person or thing that turns out to be very advantageous.
Mit dieser neuen Mitarbeiterin haben wir einen wirklich guten Fang gemacht.
With this new employee, we have made a really good catch.
When 'Fang' means the prey of fishermen or hunters, it is often combined with adjectives such as 'reich' (rich/plentiful), 'gut' (good), or 'mager' (meager). Typical verbs are 'einen Fang machen' (to make a catch) or 'den Fang verkaufen' (to sell the catch).
Nach dem Sturm war der Fang der Fischer sehr mager.
After the storm, the fishermen's catch was very meager.
The meaning 'Eckzahn eines Raubtieres' (canine tooth of a predator) is technical language from hunting or biology. You can recognize it by the fact that it is about animals such as wolves or wild boars, and the plural 'die Fänge' (fangs) is often used.
Der Tiger zeigte seine langen, spitzen Fänge.
The tiger showed its long, sharp fangs.
game / section
'die Partie' and 'der Fang' can both refer to a person who is considered especially advantageous or attractive. With 'eine gute Partie' (a good match), you mainly think of a potential spouse or long-term partner, often with a view to money, status, or good character. 'Ein guter Fang' (a good catch) sounds more colloquial and can also be used for a valuable thing or a good opportunity. Learners may confuse the words because both sound positive. In practice: for a potential spouse, 'eine gute Partie' fits; for something you have successfully obtained, 'ein guter Fang' often fits.

Mit der neuen Wohnung in dieser Lage hat Lisa wirklich einen guten Fang gemacht.
With the new apartment in that location, Lisa really made a good catch.
Viele fanden, dass der junge Arzt eine gute Partie war.
Many people thought the young doctor was a good match.