_____~긴 뭐가 ______ = Denying what somebody says

가: 아니야, 그 때문이 아니야! = No it’s not because of that.
나: 아니긴 뭐가 아니야! = What do you mean no! 

가: 저 여자 좀 봐! 저렇게 예쁜 여자가 있었나? = Look at that woman. There are women that pretty!?
나: 예쁘긴 뭐가 예뻐? = She’s not pretty.

가: 힘들어 죽겠어 ㅜㅠㅠ = It’s so hard I’m gunna die.
나: 힘들긴 뭐가 힘들어! 힘내라! = It’s not hard! Keep going! 

~기는(요) =  Used to emphatically disagree with a previous statement

(shows modesty in reaction to a compliment) x

이 책을 다 읽었어요? = Have you finished reading the book?
끝나기는요? 책을 읽는 것을 시작하지 않았어요! = Finished? I haven’t started reading it!

지금 너무 춥죠? = It’s really cold now, right?
춥기는요. 다음 달에 기온이 더 낮아질 거예요. = Cold?! Next month the temperature will get lower.

한국말을 잘하시네요. = You speak Korean really well.
잘하기는요. 아직도 실수를 많이 해요. = Well? I still make lots of mistakes.

옷을 잘 입어요. = You dress very well.
잘 입기는요. = Dress well? Oh not at all… (blushing)

~기는(요) is sometimes simply the ~기 nominalizer followed by topic particle 는 i.e. speaking of… when it comes to… as for...

칠칠맞기는 = Talk about sloppy.
살아 있기는 살아 있을까? = But do you suppose they’re really alive?

Denying/defending

그렇지 않아요 = That’s not how it is, It’s not like that.
그게 아니고요 = That’s not it
그건 아닌 것 같아요 = I don’t think that’s it/how it is/true
제 말은 그게 아니고요 =That’s not what I’m saying/I mean
절대 그럴 리가 없어요 = That’s impossible, That can’t be true.
말도 안 돼요 = That’s not even possible,

It doesn’t even make sense.
그런 뜻으로 한 말이 아니에요 = I didn’t mean that by what I said, What I said doesn’t mean that.
뭔가 잘못 아셨을 거예요 = You must be mistaken about something.
분명 뭔가 오해가 있는 것 같아요 = Clearly there must be a misunderstanding.
꼭 그런 건 아니고요 = It’s not exactly like that.
그렇다고 보기는 힘들죠 = It is hard to say that’s the case.
아직 확실히 정해진 건 아니에요 = It’s not decided for sure yet.