Why were the British willing to bet the Oregon Territory with the Americans

To take advantage of the situation in the America.

Explanation:

The British willing to bet the Oregon Territory with the Americans because at that time American Mexican war started and Americans can't afford two war at the same time and the Americans also knew that Britain has a powerful military. So the Britain take advantage of this situation and made a bet on Oregon territory and the Americans have no choice to accept it in order to avoid two wars.

small european communities with shared traditions consolidated into large new states (apex)

the anwser is a) to prevent military conflict