[Spicymoose]

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Please don't take this as a knock on your play, but I think it is a tough situation. Here is the dilemma, call a check raise on the river. You have represented at least top pair...and could very easily have raised with KJ pre flop...
That being said...what do you see the majority of the time Head UP on a check raise at the river???
Usually Either Nuts or Bust....Never a I think I just made the best two pair hand or set...Unless set is nuts...
I would fold the river, however, since the hand is won or lost already, and I really am not sure where I am at on river, I just check behind and show my hand down.
He could have AK...AQ...or KJ...or taking a shot with KT or even JT hoping you fold KJ which is unlikely.
Fold River, it is correct more often than not, and next time check river with top pair second best kicker in a connected broadway board.
Good Luck

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You really wan't him to miss his value bet on the river? That seems bad. As for whether or not to pay off the check raise... You need to be good 1 in 11, and although I do agree that he could be bluffing occasionally, I think there are far too many possibilities of hands that beat you. The only reason to call would be if you were willing to take a slight -EV hit to get a better understanding of how your opponent plays his cards. I think you might be good 1 in 20 here (random guess, I could calculate more exactly, but I think this is a fair guess), so if you are willing to pay somewhere around .5 BB, maybe a bit less, to get to know your opponent better, by all means do it. I think this is a fold though.