Final answer:
The missing section in the sentence is 'to vote.' After gaining the right to vote, African Americans began to organize politically and participate in governing the South.
Step-by-step explanation:
The missing section in the sentence is 'to vote'.
Having gained the right to vote politically and to take part in governing the South, African Americans quickly began to organize politically and to take part in governing the South.
This refers to the time after the Civil War when African Americans were granted the right to vote with the 15th Amendment. They took advantage of this right to participate in politics and hold positions of power. African Americans organized themselves effectively and elected black members of Congress, contributing to the political landscape of the South.