The statement that half of the present day United States belonged to Mexico before the Treaty of Guadalupe Hidalgo is not quite true. The land that the US took from Mexico in that treaty (after the Mexican-American War) was about half of Mexico’s territory at that time. However, it is nowhere near to one half of what is now the United States.
In 1848, the US and Mexico signed the Treaty of Guadalupe Hidalgo. This was a treaty that officially put an end to the war between the two countries that had started in 1846. The treaty gave the United States a very large area of land. This land included all of what is now California, Nevada, and Utah as well as parts of Arizona, New Mexico, Colorado and Wyoming. While this is a large area of land, about 525,000 square miles, it is not half of the land area of the present-day United States. Instead, it is only about 15%, or one-seventh, of the country as it now exists. Just the contiguous 48 states of the US have a land area of more than 3 million square miles.
The Mexican Cession did add a great deal of land to the United States, but it is not quite accurate to say that one half of what is now the US belonged to Mexico right up until 1848.