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The only time I would check/call the turn here is if BB habitually calls on the flop but loves to raise on the turn. There are a good number of players who do just that. In that case, check-calling is the superior play.

Against normal opponents though, I think betting this turn is right. In general you have a good measure of folding equity. It's very difficult for him to continue without a Q or J. He might have one, but it's more likely that he doesn't. If he does and you get raised, you have outs and can probably pull off a river checkraise for good measure.

I think you played it fine.

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Agreed.