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i think coldcalling 3bets is a bad play in general because if we do this with big pairs then we can't 4bet bluff nearly as profitably against thinking opponents (which BTN may or may not be).
in a vacuum, i think your preflop line is okay. typically, and without reads, id think c/c flop c/f turn would be your best postflop line. however this particular turn card gives you several outs versus K-K+ so i need to do a little maths to see if your line is +EV...
This is your equity versus UTG, if his range is K-K+
Board: Td 5d 8s 9d
Dead:
equity win tie pots won pots tied
Hand 0: 21.591% 21.59% 00.00% 114 0.00 { QdQh }
Hand 1: 78.409% 78.41% 00.00% 414 0.00 { KK+ }
rough EV analysis: approx. $36 in pot, risk approx. $40 to win approx. $100. therefore you would need about 40% equity to make going all in on the turn +EV. clearly the BTN would have to have a high bluffing frequency in this spot for that to be +EV (math wise, about 20% of his range would have to be air).
i don't like check/calling the turn because villain is unlikely to felt a non diamond K-K/A-A if a 4th diamond hits. villain may pay off a straight, but i doubt he would do so 100% of the time.
therefore i'd check/fold the turn without additional reads.
note: adding AKdd to villains range reduces your equity by about 3%
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