There are two similar words - 끊어지다 and 끊기다 in Korean. Both seem to be intransitive verbs and are different from 끊다.
But except that, is there any difference between them? I could not detect anything from the example sentences on the dictionary...
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They are always interchangeable because they are (absolute) synonyms.
Suffixes -이-, -히-, -리-, and -기- add passiveness to a specific group of verbs. 끊기다 can be decomposed into 끊- + -기- + -다; the passive verb 끊기다 directly relates to the transitive verb 끊다.
(Note: Suffixes -이-, -히-, -리-, -기-, -우-, -구-, and -추- add causativeness to another specific group of verbs.)
(Note 2: There is another word 끊이다. Its meanings have passiveness, but 끊이다 and 끊다 are not directly related to each other. Please consult the dictionary to figure out how different they are.)
(Note 3: I have said specific, because the use of those suffixes is limited.)
지다 of 끊어지다 is the auxiliary verb that can add passiveness to a lot of verbs. This 지다 is used in the form of -어지- (including -아지-, -워지-, and -와지-) without spacing between -어 and 지-. Although the form has changed over time, 끊어지다 has been used for at least several centuries.
(Note: I have said can, because 지다 may also be used as in "(너는) 그 상황에 웃어지더냐? (Was it possible for you to laugh / smile in that situation?)")
Therefore, 끊기다 and 끊어지다 are semantically the same words.
You might have noticed that some people use 잊혀지다 making the "double passive" (이중 피동) error. 잊혀지다 is decomposed into 잊- + -히- + -어지- + -다. 잊히다 is already a passive verb; 잊어지다 is not preferred.
A lot of people, even those who were born in Seoul, sometimes use 놀래다 instead of the verb 놀라다, although 놀래다 is a causative verb originally decomposed into 놀라- + -이- + -다. This is also a common error that they do not notice well.
It is incorrect to use 써 있다 (as in 여기에는 위험하다고 써 있다) to mean 쓰여 있다. The passive verb 쓰이다 is decomposed into 쓰- + -이- + -다.