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 rules:** The area of a square is given by $\text{side}^2$. 3. **Calculate the area of the outer square:** The outer square has side length $a + b$, so its area is $$ (a + b)^2 $$ 4. **Calculate the area of the inner square:** The inner square has side length $c$, so its area is $$ c^2 $$ 5. **Calculate the area of one right-angled triangle:** Each triangle has legs $a$ and $b$, so its area is $$ \frac{1}{2}ab $$ 6. **Express the relationship between areas:** The outer square is composed of the inner square plus four right-angled triangles, so $$ (a + b)^2 = c^2 + 4 \times \frac{1}{2}ab $$ 7. **Simplify the right side:** $$ (a + b)^2 = c^2 + 2ab $$ 8. **Expand the left side:** $$ a^2 + 2ab + b^2 = c^2 + 2ab $$ 9. **Subtract $2ab$ from both sides:** $$ a^2 + \cancel{2ab} + b^2 - \cancel{2ab} = c^2 $$ 10. **Final result:** $$ a^2 + b^2 = c^2 $$ This completes the proof of the Pythagorean Theorem using the given method.