Hello,
Preflop we could consider raising 3bb from this position.
Why do we cbet so big on the flop here?
I would cbet around $0.55 here.
As played, considering he called our big cbet I think he doesn't have air hands in his range.
So I think his range here is capped between PPs (probably 5x) and flush draws.
If we give him this range - {TT-33,A5s,A3s,65s,AcQc,KcQc,AcJc,KcJc,QcJc,AcTc,KcTc,QcTc,JcTc,Ac9c,Kc9c,Qc9c,Jc9c,Tc9c,Ac8c,Tc8c,9c8c,Ac7c,9c7c,8c7c,Ac6c,7c6c,Ac4c,Ac2c} we have around 45% equity so check/raising doesn't look like a decent option, since we are not having any fold equity, and we have less than 50% against his range.
So we are left with check/calling or betting ourselves or check/folding.
I think if we bet we could have some fold equity against 3x/smaller FD so we could have some fold equity.
If we go for the check/call we have to be sure that he's going to bet the turn as a bluff and we have to call him down.
Check/folding is a little bit weak, but without knowing exactly what to do this is always the best option.
Without inforamtion I would personally bet around 1/2 pot and fold to a raise.
As played, I think check/calling the river is fine.
We can never make him fold a made hand, but he might decide to pull up some stupid bluff with something like Ac9c.
Best,
Plamen