There are over 400 known proofs of the Pythagorean Theorem, and each proof demonstrates the validity of the theorem in a unique way. Proofs are especially helpful when the concept being proven is not obvious, as in the case of the Pythagorean Theorem. Although it has been established in the chapter “History of the Pythagorean Theorem” that the Pythagorean Theorem is true, working through the proofs in this chapter can help you gain a deeper understanding of the theorem and under what conditions it holds.
Writing proofs requires organization, and we recommend using the following format for every proof you do. For each proof, you should include a section listing the “given,” or the proof's set-up, which may involve shapes, labels, or angle measurements. Next, you should include a statement of what you are going to prove in a “prove” section. In the next section, the “proof” section, work through the proof systematically and explain each step you take. If a proof requires many steps, it may be helpful to number each step as a way to stay organized. At the end of your proof, it’s always a good idea to have a “conclusion” section that states what your proof demonstrates. We will use this format for the following proofs.