Final answer:
Mexico ceded the lands that became the Southwest Territory, through the Treaty of Guadalupe Hidalgo in 1848, after the Mexican-American War.
Step-by-step explanation:
The state that ceded the lands that made up the Southwest Territory to the United States was Mexico. This occurred through the signing of the Treaty of Guadalupe Hidalgo at the end of the Mexican-American War in 1848. The treaty confirmed the U.S. title to Texas and ceded the Alta California and New Mexico territories to the United States. The land that Mexico ceded under the treaty eventually became all or parts of the states of California, Arizona, New Mexico, Texas, Utah, Nevada, Colorado, Wyoming, and Kansas, leading to a significant territorial expansion of the United States and furthering the concept of Manifest Destiny.