Baja California was never taken from Mexico. The territory that was lost however, included California. The Treaty of Guadalupe Hidalgo designated which extension of territory was given to the US after the Mexican-American War (1846-1848).
The US gained control of the Utah Territory based on the treaty with Mexico that ended the Mexican War
The Treaty of Guadalupe Hidalgo ending the Mexican American War.
They added former Mexican territory to the United States.
The Battle of Fallen Timbers and Pinckney's Treaty The Louisiana Purchase The Mexican-American War
The Mexican States of Alta California and Nuevo Mexico.
The Mexican-American War ended with the signing of the Treaty of Guadalupe Hidalgo. This treaty gave a lot of Southwestern territory to the United States. The United States gave $15 million to Mexico, and $3.25 million in debt owned to Mexican citizens.
The Southwest, including California, was taken "by force of arms" from Mexico, during the Mexican War.
The name of the treaty of Louisiana territory is The Louisiana Purchase.
peace treaty
The treaty allowed the United States to purchase the territory from Mexico.
What officially ended the Mexican War was Treaty of Guadalupe Hidalgo. I'm sorry but, I do not know the answer for the second question. I hoped this helped!