1. **State the problem:** Prove the Pythagorean Theorem using the method of arranging four right-angled triangles around a square of side length $c$. 2. **Formula and explanation:** The outer square formed has side length $a+b$, so its area is $$ (a+b)^2 $$ 3. The inner square has side length $c$, so its area is $$ c^2 $$ 4. Each right-angled triangle has area $$ \frac{1}{2}ab $$ 5. The outer square's area equals the inner square's area plus the area of the four triangles: $$ (a+b)^2 = c^2 + 4 \times \frac{1}{2}ab $$ 6. Simplify the right side: $$ (a+b)^2 = c^2 + 2ab $$ 7. Expand the left side: $$ a^2 + 2ab + b^2 = c^2 + 2ab $$ 8. Subtract $2ab$ from both sides: $$ a^2 + \cancel{2ab} + b^2 - \cancel{2ab} = c^2 $$ 9. This simplifies to: $$ a^2 + b^2 = c^2 $$ **Final answer:** The Pythagorean Theorem is proved: $$ a^2 + b^2 = c^2 $$