Final answer:
The Democratic Party gained control of the presidency in 1912 when Woodrow Wilson won the election, benefiting from a split within the Republican Party, which had been dominant during the Fourth Party System.
Step-by-step explanation:
The political party that gained control of the presidency for only the second time in 50 years in 1912, when the other major national party split, was the Democratic Party. This occurred during the Fourth Party System, a period characterized by Republican Party dominance. The split within the Republican Party occurred between the incumbent President William Taft and former President Theodore Roosevelt, who led the Progressive Party, also known as the Bull Moose Party, after failing to secure the Republican nomination.
Woodrow Wilson, the Democratic candidate, took advantage of the divided Republican vote to win the presidency. Wilson won with just 42 percent of the popular vote, due to the rift in the Republican Party, which also led to Democrats gaining control of the Senate while already holding the House majority.