Taming of the Shrew

Act III Act I Scene: A room in Baptista's House
What is the quarrel between Lucentio (as Cambio) and Hortensio (as Litio)?
They were arguing over who will teach Bianca, and who will start teaching first.
How is it settled?
When Bianca pipes in and tells them who is teaching first.
When Lucentio reveals his real identity to Bianca, how does she feel about him?
She doesn't trust him. But, she agrees to keep his secret so she must like him because she didn't send him away.
Has either made any progress during this scene?
Lucentio reveals himself and she's going to keep his secret
Compare it with the courtship of Kate and Petruchio. Do you think Petruchio does a more interesting job? Do you think his courtship is more intriguing? Why?
Yes, he does a more interesting job, and it is more intriguing because how he handles and talks to her is romantic, yet clever and somewhat harsh.
What ends the scene with Bianca and her two suitors?
When the messenger(servant) comes in and tells her she must prepare for her sister's wedding.
What warning does Hortensio give to Bianca before she leaves?
"Show pity or I die." (line 77)
Do you think Bianca would be likely to take Hortensio's threat very seriously?
No, because she's already learned it and he is a babbling fool.