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mexico lost half its territory to which country? why did mexico lose so much of their land?

The correct answer is - The United States.

Mexico controlled the western, southwestern, and the westernmost part of the Midwest of what is now the United States. But the United States were growing stronger, and the Spanish support and power was declining, putting Mexico in a position to control very large territory which was not really able to properly defend. The United States had already started with expansionist politics, and once they saw that there's a good opportunity in taking over some of the Mexican territories, they did so, and managed to conquer them, incorporated them into their territory, and gained and even better strategical position.