1. Stating the problem: Create a table with two columns. The first column contains statements of theorems. The second column contains the reasoning/proof behind each theorem. 2. Example: | Statement of Theorem | Reasoning/Proof | |----------------------|-----------------| | Pythagorean Theorem: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: $$a^2 + b^2 = c^2$$ | Using the properties of Euclidean geometry and similar triangles, one can prove that the areas relate this way. | 3. Another example: | Statement of Theorem | Reasoning/Proof | | Fundamental Theorem of Algebra: Every non-constant polynomial equation with complex coefficients has at least one complex root. | This is proven using complex analysis and properties of continuous functions, such as Liouville's theorem or Rouche's theorem. | 4. General Approach: - For each theorem, write its clear, concise statement in the first column. - In the second column, provide a brief but solid proof or reasoning explaining why the theorem holds. - The table format allows structured and easy comparison. No specific theorems are provided to populate a full table here, but the user can create such tables following this template.