The defeat of Mexico after the Mexican American War firmly established the Rio Grande River as the border between Mexico and Texas.

After the Mexican American War, the defeat of Mexico resulted in the establishment of the Rio Grande River as the border between Mexico and Texas. This process can be understood through the following steps:

1. Mexican American War: The war between Mexico and the United States took place from 1846 to 1848. The conflict arose due to territorial disputes, with the United States seeking to expand its territory westward.

2. Defeat of Mexico: The Mexican forces were ultimately defeated by the United States, leading to the signing of the Treaty of Guadalupe Hidalgo in 1848. This treaty marked the end of the war and established the terms of peace between the two nations.

3. Border Establishment: As part of the Treaty of Guadalupe Hidalgo, Mexico ceded a significant portion of its territory to the United States. This included the recognition of the Rio Grande River as the official border between Mexico and Texas.

4. Significance of the Rio Grande River: The Rio Grande River, also known as Río Bravo del Norte, is a natural boundary that stretches approximately 1,900 miles (3,100 kilometers) from the Rocky Mountains to the Gulf of Mexico. Its establishment as the border between Mexico and Texas was a significant outcome of the Mexican American War.

5. Legal and Political Implications: The recognition of the Rio Grande River as the border had legal and political implications for both Mexico and Texas. It solidified the territorial claims of the United States and established a clear demarcation between the two nations.

Overall, the defeat of Mexico in the Mexican American War led to the establishment of the Rio Grande River as the border between Mexico and Texas, a significant outcome that continues to shape the relationship between these two countries.