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Falconi had her best career result in getting to the semifinals at this event last year, and found herself down a set but up 5-3 in the second against the fourth-seeded King. It eventually went to a tie-break, where Falconi went up 5-4 but once again King took back the momentum and closed out a 62 76(5) victory.
"We both changed our game styles a little bit during that match," King said. "In the beginning she was putting the ball in the court, waiting for me to miss, and I was missing. I was kind of seeing how she was playing, getting used to it, putting her on the defensive. She started playing much more aggressive in the second set, which forced me to play more aggressive also.
"Overall we played at a good level. I thought it was a very good match. I am really happy I ended up pulling ahead in the second set."
In her second quarterfinal of the year after Pattaya City, King will play another American - Coco Vandeweghe. The seventh-seeded 20-year-old cruised past Aravane Rezai, 62 62, her third win in four meetings with the Frenchwoman.
"I have played Rezai before, and I feel like those two times I was a totally different player than I was today," Vandeweghe said. "I went in with the game plan that if I played my patterns and my game, I was going to beat her. That's what I did, and I felt even more comfortable on the court today than I did in my first round. I am happy with the result."
Vandeweghe talked about how making her first career final at the Bank of the West Classic in Stanford last month affected her game.
"I won a lot of matches on grass before Stanford and that totally boosted my confidence going into that tournament," Vandeweghe said. "Beating the players that I beat in Stanford helped me feel that all the work I put in practice and on the court is really showing."
Top-seeded Anastasia Pavlyuchenkova was a 62 26 63 winner over Melinda Czink, her second triumph in three encounters with the Hungarian - with all three having gone three sets. After not making a quarterfinal all year, Pavlyuchenkova has now gotten that far in three of her last four events.
The surprise of the day saw Eugénie Bouchard record a 16 60 63 defeat of No.8 seed Olga Govortsova. The 18-year-old Canadian was down 3-0 in the third set before reeling off six straight games. She now has three career WTA match wins, and all of them have come at this event - one last year and two so far this year. Next up for her is another teen - 19-year-old Sloane Stephens.
Setting up an all-Slovakian quarterfinal were Magdalena Rybarikova and Jana Cepelova. Rybarikova rallied from 5-2 down in the second set for a 63 75 win over Eleni Daniilidou. Cepelova came from 3-0 down in the second to beat Edina Gallovits-Hall, 63 64, and advance to a final eight for the first time.