To my ears, ㄱ is pronounced as g or k depending on the word.
- 강남 -> k
- 내과 -> k
- 아기 -> g
Is there any simple rule-of-the-thumb?
Note. This is not a duplicate of this as that thread discussed romanization despite asking for pronunciation too.
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The pronunciation doesn't differ depending on the word (normally), but depending on its environment - where it appears in the syllable or relative to other sounds. This is true for any sound in any language (for example, the English 'k' phoneme is aspirated at the beginning of a word like kill but unaspirated/less aspirated after an 's' like in skill); in phonology, we say that ㄱ represents one phoneme (with exceptions like 내과 - this is just a spelling irregularity where ㄱ actually represents the ㄲ phoneme) but there are various allophones of the phoneme depending on the environment it is found in. Understanding the concept of allophones of a phoneme is very useful for learning pronunciation in another language.
So in the example of ㄱ, it can be pronounced several ways:
In the case of 내과 (and some other words ending in 과 or 권 like 외과, 치과, 여권, etc.), it is a rare case where the spelling does not match the pronunciation. These are spelled with 과 because the underlying 한자 in the word 科 is pronounced 과 in many words like 교과서 and 과목. However, at some point in time there was a change in the pronunciation that has not been reflected in the spelling. It's best to memorize those cases.