At the beginning of Act III, Scene 1, Tybalt approaches Mercutio because...

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he wants to make peace with the Montagues.
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they are kinsmen.
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Mercutio yelled to him from across the road.
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he recognizes Mercutio as a friend of Romeo.

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At the beginning of Act III, Scene 1, Tybalt approaches Mercutio because he recognizes Mercutio as a friend of Romeo. Tybalt is looking to confront Romeo, and since Mercutio is associated with him, he targets Mercutio instead. This confrontation ultimately leads to a tragic series of events.