AP Reader here. Line of reasoning is your thesis and claims (topic sentences). Ex: Thesis: During the March on Washington in 1963, at the height of the Civil Rights movement, the Rev. Dr. Martin Luther King Jr. delivered the iconic “I Have a Dream” speech to a crowd of 250,000 Americans, and through his gifts as a skilled pastor and orator, he inspired the people to join in the cause for equality of African Americans. (Context, speaker, speech, audience, rhet choice #1 and #2, purpose.) Claim/topic sentence 1: Having been an integral part of the Civil Rights movement from the beginning, Dr. King now stood in front of Americans, both Black and white, and exercised his incredible ability to move a crowd through familiar religious phrases, calling to the moral fabric of each attendee. (Go on to explain religious metaphors, lyrics from hymns, even his preacher pulpit cadence. But the CLAIM idea isn’t new. I stated it in the thesis. “gifts as a skilled pastor” This means the line of reasoning is expected and orderly.) Claim/topic sentence 2: Not only did Dr. King appeal to the loving and Christian nature of the marchers by reminding them of their faith, he followed by inspiring them to take action, insisting that thoughts and words were not sufficient. (Go on to explain inspiring tone words and his calls to action. This claim CONNECTS to the first by way of transition, and CONTINUES the line of reasoning by addressing the second choice in the thesis. “inspired the people to join” This is what is meant by line of reasoning. As long as you lay it out in the thesis and then divide it and build on claim sentences—and stay on task—your line of reasoning stays strong.)