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Brânquia vs. guelra

While both brânquia and guelra translate to "gill" in English, the difference lies primarily in the register and context. Brânquia is a scientific term used in biology, whereas guelra is the common, everyday term used in fishing and cooking.

Brânquia

B2
This is the anatomical and scientific term for the respiratory organ. You should use brânquia when discussing biology, zoology, or the specific mechanics of how aquatic animals extract oxygen from water.
O biólogo explicou como funciona a brânquia do tubarão.
(The biologist explained how the shark's gill works.)
As brânquias filtram o oxigênio dissolvido na água.
(The gills filter the oxygen dissolved in the water.)
A poluição pode causar danos severos nas brânquias dos peixes.
(Pollution can cause severe damage to the fish's gills.)
Alguns anfíbios mantêm suas brânquias durante a fase adulta.
(Some amphibians keep their gills during the adult phase.)
O parasita se alojou na brânquia direita do animal.
(The parasite lodged itself in the animal's right gill.)

Guelra

C1
This is the popular term, often referring to the red slit or the flap behind the head of the fish. Guelra is used by fishermen, cooks, and in fish markets. It can also be used metaphorically in idioms to describe someone with energy or gumption.
Para saber se o peixe está fresco, verifique se a guelra está bem vermelha.
(To know if the fish is fresh, check if the gill is very red.)
O pescador segurou o peixe grande pelas guelras.
(The fisherman held the large fish by the gills.)
Você deve limpar as guelras antes de assar o peixe.
(You must clean the gills before baking the fish.)
A faca deve passar logo atrás da guelra.
(The knife should pass right behind the gill.)
Aquele rapaz tem muita guelra para os negócios.
(That young man has a lot of gumption for business.)

Summary

In short, use brânquia for scientific precision and biological descriptions. Use guelra in everyday life, especially when handling fish in a market or kitchen, or when referring to the external visible part of the organ.