1. **State the problem:** Given a right triangle $\triangle FKJ$ with right angle at vertex $K$, prove that $a^2 + e^2 = d^2$ where $a = FK$, $e = KJ$, and $d = FJ$ (the hypotenuse). 2. **Identify the theorem to use:** This is a classic right triangle problem where the Pythagorean theorem applies. The Pythagorean theorem states: $$\text{In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:}$$ $$c^2 = a^2 + b^2$$ where $c$ is the hypotenuse. 3. **Apply the theorem to the triangle:** Here, the hypotenuse is $d = FJ$, and the legs are $a = FK$ and $e = KJ$. So by the Pythagorean theorem: $$d^2 = a^2 + e^2$$ 4. **Conclusion:** Thus, the statement $a^2 + e^2 = d^2$ is justified by the **Pythagorean theorem** because $\triangle FKJ$ is a right triangle with right angle at $K$. **Final answer:** $$a^2 + e^2 = d^2$$ This completes the proof using the Pythagorean theorem.