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I think check-folding would have been the best play, since I probably wouldn't have got the turn card cheap enough to call (you noticed I had only the backdoor flush draw, right?).

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Well, the idea would be to pick up a flush draw on the turn and then semi-bluff all in (check-raising all-in would be ideal, but it doesn't seem like stacks are deep enough for that). Don't get me wrong, you will still be all-in with a dog a lot of the time, but this will give you at least some fold equity that you don't have on the flop. Even better, your check call might confuse him into checking behind. This also lets him bluff at you with three outs if he's got AQ, AJ, KQ, etc.