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Hand 1 is fine. Although your hand itself (TP + NFD) is a little overrated (I would rather have second pair + NFD) because hitting trips probably doesn't really change anything. For example I wouldn't be as eager to get it all in on the flop with A2c on a Ad Jc 8c flop as I would with A2c on a 2d Tc 9c flop, because often times in the first hand you're going to be basically on a glorified flush draw, your kicker may not even be out if they have aces up, so you basically have 9 outs a lot of the time. In the second flop we have a lot of outs against something like T9, any two, any ace, any club. There is a big difference between those two hands. In either event hand 1 is just fine. There are times when he is on a total bluff or has a lower flush draw, not to mention you are only a small dog to a better ace. It's not like a massive +EV situation, but it's a fine time to get your chips in when the pot is heads up. This situation plays much differently multiway.
Hand 2: Very sticky spot, I'd raise bigger on the flop. As is stands you can't do anything but c/f the turn. You're behind pretty much everything on that board. I have a feeling you were probably beat all along so the turn saved you money (assuming you folded).
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