How did Mexican lands in the West become part of the United States?

The Treaty of Guadalupe Hidalgo is the short answer.

Explanation: After Mexico lost the Mexican-American War, the US paid Mexico $15 Million dollars to cede the lands in the West. This is also called the Mexican Cession

frederick william of brandenburg-prussia trained his son, frederick, to be a hardy soldier. but frederick disliked rifle shooting and horseback riding. unlike his father, he was interested in french literature. he enjoyed poetry and music and, like some others of the enlightenment, he scoffed at religion. his father thought him soft, frivolous and unmindful of his duties. and when frederick refused to put his signature on a right to succession, for a few moments his father strangled him.