How does Romeo use compliments to win Juliet over?
When Romeo and Juliet meet at the Capulets' big party, one of the first things he compares her to is a "holy shrine," suggesting he is unworthy even to touch her, but that his lips are like "blushing pilgrims" that might help smooth the "profane" touch of his hands with a kiss (I.5.104-106). Romeo implies Juliet is divine, imbued with some force that is higher than human, and that he feels undeserving of her. Along these same lines, Romeo later calls Juliet a "saint" (I.5.114). Later, when Romeo speaks to Juliet on her balcony, he again calls her a "dear saint" (II.2.60). When Juliet expresses her fear that her kinsmen will find Romeo and kill him, he says,
Alack, there lies more peril in thine eye
Than twenty of their swords.
Look thou but sweet,
And I am proof against their enmity (2.2.76-78).
In other words, then, Romeo assures Juliet that she has far more power over him than any weapons do. Romeo claims that, if Juliet cares about him, he cannot be harmed by her relatives. Further, he says that "[His] life were better ended by their hate / Than death prorogued, wanting of [her] love" (II.2.82-83). Romeo says he doesn't even really consider his life worth living—that he would rather die—if he doesn't have Juliet's love. I'm not sure that Romeo's compliments are solely responsible for winning Juliet over, but it can't hurt!