What happens
Kate brutally interrogates Bianca about her suitors, then strikes her. Baptista scolds Kate and sends Bianca away. Petruchio arrives with tutors for Bianca, immediately declares his intention to marry Kate, and engages her in a battle of wit and insult. Despite Kate's hostility, Petruchio announces their engagement, claiming she loves him deeply. Baptista, amazed, agrees to the match for Sunday.
Why it matters
This scene establishes the central dynamic of the play: Kate's verbal aggression as both shield and weapon. Her interrogation of Bianca—demanding to know whom she loves—masks jealousy beneath cruelty. She strikes Bianca, revealing that her shrewishness is performative, a response to her position as the 'unwanted' elder daughter. When Baptista dismisses her, calling her a 'devilish spirit,' Kate's rage becomes intelligible: she is trapped by her father's authority and society's expectations. Her final soliloquy about dancing barefoot at Bianca's wedding and leading apes in hell distills her predicament—unmarried women are social failures, yet she actively resists the remedy.
Petruchio's entrance marks a turning point. He arrives as a fortune hunter, explicitly seeking a wealthy wife 'any woman.' But his sparring with Kate reveals something unexpected: he meets her aggression not with domination but with mirroring and wit. When she calls him a 'moveable' (a piece of furniture), he responds with sexual innuendo. When she insults his youth, he compliments her walking. This is not taming through force but through a game of contradiction where both players seem to understand the rules. His claim that she loves him 'ten times more than e'er I did' is absurd—they've just met—yet it establishes his method: he will reshape reality through sheer assertion, and Kate will be forced to decide whether resistance or compliance is the better strategy.