• 15-year-old through to Wimbledon third round after 6-3, 6-3 win • Gauff is set to face Polona Hercog in round three

We had better get used to this. The 15-year-old American Cori Gauff produced a performance oozing with confidence and maturity as she backed up her win over Venus Williams with an even better display on Wednesday to reach the third round at Wimbledon.

Avoiding the let-down that so often afflicts giantkillers, she beat Magdalena Rybarikova of Slovakia, a semi-finalist here two years ago, 6-3, 6-3. Upsetting Williams might have been a surprise but this was seriously impressive.

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Returning to No 1 Court, the scene of her win over five-times winner Williams, Gauff danced her way around the court in a manner reminiscent of Venus at her best while even her “come on” had a ring of Serena Williams. Her movement was electric, her serve huge, her ground strokes crisp and her calmness was perhaps the most impressive thing of all.

Patrick Mouratoglou, the long-time coach of Serena Williams, who has been working with Gauff for a number of years, had been concerned that the teenager would find Rybarikova’s changes of pace and all-court game hard to deal with. He need not have worried. Gauff rushed Rybarikova with the quality of her ball-striking and the Slovakian had no answer.

With the new roof closed, seemingly to ensure the flow would not be broken if it had to be shut midway through the match, Gauff began confidently and after a big return set up three break points in the sixth game, broke to lead 4-2 when Rybarikova missed a backhand.

Gauff marched around the court between points as if she had been doing it for years at this level and after wrapping up the first set, she broke early on for a 2-1 lead in the second set. Trailing 3-1 and 15-40, Rybarikova saved two break points to stay in touch but Gauff stayed solid on her serve and broke again to clinch victory. She will play Polona Hercog of Slovenia, who ousted 17th seed Madison Keys, 6-2, 6-4, in the next round.

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Two former world No 1s will meet in the third round after Simona Halep and Victoria Azarenka came through in contrasting fashion. Seventh seed Halep needed three sets to overcome another Romanian, Mihaela Buzarnescu 6-3, 4-6, 6-2 while Azarenka dropped the first two games before winning the next 12 as she hammered Ajla Tomljanovic of Australia 6-2, 6-0.

Azarenka, unseeded this year as she continues to climb back up the rankings, then won another nine straight games as she and world No 1 Ash Barty won a doubles match, making it a run of 21 straight games. “The second one was with help,” she said, referring to an injury to her opponents. “The first one, no help. First 12 with no help, but then it was help. It was cool.”

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Halep said she was looking forward to another tough battle with Azarenka. “We played I think two years ago here,” she said. “It’s going to be a big challenge for me, tough one, because she’s playing not very flat but she’s very aggressive. So on grass it’s not that easy. I am confident I have my chance. I want to win badly every match I play.”

Another former world No 1, Caroline Wozniacki, joined them in the third round with a hard-fought 7-6, 6-3 victory over Veronika Kudermetova, gaining revenge for her defeat by the Russian in the first round of the French Open last month. Wozniacki, who announced last October that she is battling with rheumatoid arthritis, made just six unforced errors and didn’t face a single break point. “Very happy with that. She’s a tough opponent, she was serving really big. It was nice to get that match back. I feel like I’ve been playing pretty well, my body’s healthy. I hope I can play at least a few more matches here.”

Wozniacki will play Zhang Shuai of China in the third round.

• This article was amended on 4 July 2019 because Polona Hercog is from Slovenia, not Poland as an earlier version said.