Final answer:
Mercutio is the character in 'Romeo and Juliet' who says 'A plague on both your houses' after being mortally wounded in a duel, cursing the two feuding families.
Step-by-step explanation:
In the Shakespeare play Romeo and Juliet, the character who says, "A plague on both your houses" after losing a sword duel is Mercutio. This line is uttered in a moment of anger and frustration over the feud between the Montagues and the Capulets, which has caused his own mortal wound. Mercutio's curse on both families foreshadows the tragic events that follow and highlights the senseless nature of the conflict that ultimately leads to the downfall of the young lovers, Romeo and Juliet.