Final answer:
The annexation of Texas by the United States was the significant event during Sam Houston's presidency that impacted the relationship between Texas and the U.S. It furthered the U.S.'s westward expansion and led to the Mexican-American War due to boundary disputes.
Step-by-step explanation:
During Sam Houston's presidency of the Lone Star Republic, a significant event that took place impacting the relationship between Texas and the United States was the annexation of Texas by the United States.
Between 1836 and 1846, Texas remained an independent nation, attracting thousands of American immigrants and debate amongst expansionists in the U.S. about the imminent annexation of the Republic to the Union. This ambition realized when Texas was officially annexed to the United States, a move that aligned with the beliefs of those who wished to expand both the nation and the institution of slavery.
It was not only a substantial event in terms of geopolitics but also in signaling the United States' westward expansion under the banner of Manifest Destiny. The annexation exacerbated tensions with Mexico, leading to the Mexican-American War, primarily over the disputed territorial boundaries.