Final answer:
Missouri is the state that lies to the southeast of Illinois, across from the Mississippi River and is separated from Nebraska and Kansas by the Missouri River, which also features the Osage River within its boundaries.
Step-by-step explanation:
The state that is separated from Illinois by the Mississippi River to the southeast, from Nebraska and Kansas by the Missouri River, and also has the Osage River within its boundaries, is Missouri. The confusion regarding the boundary between the Mississippi River and the 49th parallel stems from misperceptions about the river's source, which was believed to lie further north than it does. The significance of the Missouri and Mississippi Rivers has been felt throughout history, affecting territorial debates such as the conflict over whether the Missouri Territory would enter the Union as a slave or free state. The exploration of the Missouri River was also a crucial aspect of early American expansionism and commerce, seeking to find a water route across the continent.