The areas of California and Arizona were acquired by the US through the Treaty of Guadalupe Hidalgo.
After the Mexican-American War, in which the United States won, Mexico and the US ended the war with the Treaty of Guadalupe Hidalgo. This treaty gave America the territory known as the Mexican Cession. This territory included modern day California, Arizona, New Mexico, Colorado, and Utah. This greatly increased the size of the United States and greatly decreased the size of Mexico when it was agreed upon in 1848.