Final answer:
Bosnia and Herzegovina's annexation by Austria-Hungary in 1908 started the struggle with Serbia, leading to heightened tensions and culminating in the assassination of Archduke Franz Ferdinand in 1914, which ultimately triggered World War I.
Step-by-step explanation:
The country that was annexed and sparked the struggle between Austria-Hungary and Serbia was Bosnia and Herzegovina. The Austro-Hungarian Empire formally took control of Bosnia and Herzegovina in 1908, which had a significant Serb population. This action worsened relations with Serbia as the Serbs had nationalistic aspirations for their own expansion, seeing Bosnia as a part of a future larger Serbian state. When the Archduke Franz Ferdinand was assassinated by a Serbian nationalist in 1914, Austria-Hungary saw this as a direct challenge and an opportunity to suppress Serbian nationalism once and for all.
Despite Serbia's attempts to avoid conflict, they could not allow Austrian intervention to the degree demanded without sacrificing their sovereignty. Serbia's refusal to fully comply with Austria-Hungary's ultimatum led Austria-Hungary to declare war, with the backing of Germany. This series of events quickly escalated into what became known as World War I, with alliances drawing other major powers into the conflict. Russia, allied with Serbia, entered the war against Austria-Hungary, followed by Germany's war declarations against Russia and France, and eventually seeing the entire continent of Europe plunging into war.