Her behavior is especially bad around Bianca.
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Shakespeare doesn't seem to give us much to gon here. Perhaps one way to explain Kate's behavior would be to connect it to the theme of illusion. Shakespeare seems to deliberately exaggerate the opposite characteristics of Kate and Bianca. The men are creating illusions be disguising themselves in order to woo Bianca while Kate disguises her identity with shrewish behavior in order to drive away those men who come calling because her father is a wealthy man. She seems to submit to traditional ideas of courtship and marriage after Petruchio's persistence but her final speech is filled with irony which seems to indicate that she may yet have the upper hand.