The England captain Harry Kane scored a hat-trick, with two of the goals in second-half stoppage time, to give Bayern Munich a 3-0 victory over Augsburg on Friday. The win helped the hosts open up an eight-point gap at the top of the Bundesliga.

Bayern, who take on Paris Saint-Germain in the Champions League next week, are undefeated in the league and have 29 points after their fifth consecutive league game without conceding. Second-placed RB Leipzig, who travel to Hoffenheim on Saturday, are on 21 points.

“We said at half-time we should just keep going and create chances,” said Kane, who has 14 league goals this season. “Thankfully we got the penalty and from there the game opened up.”

Kane now has 50 league goals for Bayern in 43 Bundesliga games, reaching that milestone faster than any other player in the league’s history. “We have a big week ahead with PSG in the Champions League, then Dortmund, then Leverkusen [in the DFB-Pokal],” said Kane. “We’re feeling good and should just keep the momentum up.”

Augsburg had Nediljko Labrovic to thank for a goalless first half, with the keeper saving efforts from Kane, Jamal Musiala and Leon Goretzka. Labrovic then denied Michael Olise after half-time but could do nothing to prevent Kane from scoring a well-taken 63rd-minute penalty. The striker was then awarded another penalty after a lengthy VAR review in stoppage time with Augsburg’s Keven Schlotterbeck sent off for tripping him in front of goal. Kane again beat Labrovic from the spot before completing his hat-trick seconds later with a fine touch and header.

On Tuesday, Bayern host Paris Saint-Germain, who remain top of Ligue 1 after defeating 10th-placed Toulouse 3-0. PSG remain unbeaten domestically after goals by João Neves, Lucas Beraldo and Vitinha. Neves put the hosts in front in the 35th minute with a volley from the edge of the area, netting a neat weighted pass from Achraf Hakimi low inside the left-hand post. Beraldo doubled the lead late in the second half, scoring from a rebound, before substitute Vitinha sealed the win in stoppage time.

PSG lead the French top flight with 32 points, six ahead of second-placed Monaco, who beat Brest 3-2 earlier on Friday. Goals from Maghnes Akliouche and Aleksandr Golovin put them 2-0 up inside 25 minutes, before Abdallah Sima brought the visitors back into it after the break. Akliouche’s second made it 3-1 in the first minute of added time before Ludovic Ajorque produced a nervy last few seconds with a goal at the death.

Two goals in four second-half minutes earned Getafe a 2-0 win over Real Valladolid in La Liga. Alvaro Rodríguez scored in the 70th minute before Allan Nyom killed the game to help the hosts move clear of the relegation zone and leave Valladolid second from bottom in Spain.