Final answer:
Part of Texas' claim that the Rio Grande was the border included territory previously part of Mexico, reflecting U.S. expansionist desires and leading to the Mexican-American War and the Mexican Cession.
Step-by-step explanation:
When Texas claimed that the Rio Grande was both the southern and western border, what was part of that claim was the inclusion of a substantial amount of territory that had previously been considered part of Mexico. The annexation of Texas into the United States in 1845 led to tension with Mexico over the boundary, with the U.S. arguing for the Rio Grande and Mexico maintaining the Nueces River as the long-established border.
This dispute was rooted in the expansionist desires of the United States, exemplified by emissary John Slidell's mission to Mexico with the secret charge of acquiring additional territory for the U.S., which contributed to provoking the Mexican-American War. The war ultimately resulted in the United States acquiring large territories including New Mexico and California and establishing the Rio Grande as the U.S.-Mexico border following the Mexican Cession.