Checking is not an option with that board.
You have to bet enough to make any draw call with incorrect odds
, that's all you have to do. This is not a bad situation to be in, keep in mind you also have a 33% chance to make a full house or better from flop to river. Being that the pot is $52, you'd have to bet at least around $35/$40, and actually hope to get a caller. The only problem is that someone might have a double draw, but you can't be excessively worried about it because it's just about impossible to deny correct odds to a hand like that on the flop (ex. Ad9d).
If the turn is a brick, the same thought process applies, make draws call incorrectly, although if they both call your flop bet (which I doubt but you never know) the scenario gets a bit more complicated, because you have ~$90 left if you bet $40 and pot is $172, which brings another question: why did you raise 8.5 BB with 77 PF, also considering you're the button?