A/V + 은/는 척하다 is used to show that someone is pretending to V or be A, or they're acting as though they are A/V-ing:
To pretend to A/V: 은/는 척하다
어머니께 그에게 설거지를 하라고 말했을 때 잠을 자는 척했어요.
He pretended to sleep when his mother told him to wash the dishes.
여자는 남자친구와 헤어졌을 때 신경 안 쓰**는 척 행동했어요.
**When she split up with her boyfriend she acted as though she didn't care.
아**는 척 하지 마세요.
**Don't be a know-it-all.
그들을 들**은 척했어요.
**I pretended I was listening to them.
그 음식을 먹고 맛 있는척 했지만 사실 진짜 맛이 없었어요.
I pretended to like the food but really it was awful.
유명한척 하는 남자를 봤어요?
Did you see the guy who was pretending to be famous?
나이트에서 준영가 왜 저를 못 본척 했어요?
Why did Jun Yeong pretend not to see me in the night club?
미안하거나 슬픈 척 하지 마세요!
Don't pretend you're sorry or sad!
To brag, to boast: 잘난 척하다
그 여자는 자신의 한국어 실력을 항상 잘난 척 하고 있어요.
She is always bragging about her Korean skills.
제 친구는 완벽한 시험 점수를 받고 많은 사람들에게 잘난 척 해요.
My friend got a perfect score in the exam so he brags to everyone.