Final answer:
Two key reasons why European nations gained power in China during the 1800s were China's weakened military and political instability, paired with European countries' superior technology and military power.
Step-by-step explanation:
European nations were able to gain power in China in the 1800s mainly for two reasons: China's weakened military and political instability, and the superior technology and military power of the European countries. The First Opium War (1839-1842) significantly diminished China's strength, leading to Britain's control over Hong Kong and other key ports which emboldened other European powers. Meanwhile, China did not keep pace with European advancements in military technology, leaving them vulnerable to forced trade agreements and outright invasions.
The weakening of China was further exacerbated by their own internal problems, such as the Qing Dynasty's instability and a series of rebellions within the country. These factors, combined with aggressive European imperialism, meant that by the late 19th century, countries such as Great Britain, France, Russia, and Germany had carved out their own spheres of influence in China, securing control over trade and economics, and managing to undermine China's sovereignty.
Consequently, Europeans' effective use of military force and diplomatic maneuvering through treaties, like the Treaties of Nanking and Tianjin, further ensured their dominance and facilitated the spread of Western influence, including Christian missionaries throughout the region.