1. We are given that in triangle ABC, $a^2 + b^2 = c^2$, and there is a right triangle DEF with legs $a$ and $b$ and hypotenuse $n$. 2. Since triangle DEF is a right triangle, by the Pythagorean theorem, we know that $a^2 + b^2 = n^2$. 3. By substitution, since $a^2 + b^2 = c^2$ and $a^2 + b^2 = n^2$, it follows that $c^2 = n^2$. 4. Applying the square root property and considering the principal root, we take the square root of both sides to get $c = n$. 5. By the Side-Side-Side (SSS) congruence criterion, triangles ABC and DEF are congruent since their corresponding sides are equal. 6. Since angle $\angle F$ in triangle DEF is a right angle and triangles ABC and DEF are congruent, then angle $\angle C$ in triangle ABC is also a right angle by Corresponding Parts of Congruent Triangles are Congruent (CPCTC). 7. Therefore, triangle ABC is a right triangle as a result of having a right angle.