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While you have to respect the Ace or King on the board, you can't lay down to it every time. It demolishes your table image, and sooner or later (in all likelyhood, sooner), people will realize that you're going to let go of your cards if an overcard comes out. Sometimes you've just got to show the hand down to the river and see if you've got the best hand. Maybe you don't. But, at the very least, it helps you get a read on the other player(s) in the pot with you, and, while it might put a small dent in your chipstack for the time being, the information you've gained will definitely put you in position to win more chips.
It all comes down to the read you get off of other people. Oh, and another thing... you won't learn anything about someone's hand by checking to them and then calling their bet. If someone bets at the King, I'd consider raising to find out if he really has it. Even if he does, maybe he'll fold his K-low kicker (which he should have done preflop anyway). If you get re-raised, then you're either against a player who's not afraid to seriously bluff at the pot (most bluffers will fold to a raise, even a nominal one), or, the more likely scenario, you're in the pot against a K and you lose a little bit of money. If he merely calls your re-raise, then he's probably got a King, but not a great kicker. Consider making a stab at the pot on the turn. If he calls you, then he's probably still not sure of his kicker. If he raises you, then his not-so-great kicker might have paired, and he thinks he has the best hand with two pair. If he folds, well, congratulations... you've won yourself a pot.
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