b. Gadsden purchase. It put the Mexican / U.S. Border where it is today.
Gadsden has two d's in it.
How did California and much of the Southwest become part of the United States?
a.) Texas annexation.
b.) Gadsen Purchase.
==> c.) Mexican Cession
d.) Louisiana Purchase
All 4 involved "parts" of the Southwest, however, the most correct answer is: "C. Mexican Cession".
I hope this helps.