1. **Stating the problem:** We have a right triangle PQR with $\angle Q = 90^\circ$ and $\angle R = 2\angle P$. We want to analyze the triangle using the given angle relationships and the Pythagorean theorem. 2. **Recall the angle sum in a triangle:** The sum of angles in any triangle is $180^\circ$. Since $\angle Q = 90^\circ$, we have: $$\angle P + \angle R + 90^\circ = 180^\circ$$ which simplifies to $$\angle P + \angle R = 90^\circ$$ 3. **Use the given relation $\angle R = 2\angle P$:** Substitute into the equation: $$\angle P + 2\angle P = 90^\circ$$ $$3\angle P = 90^\circ$$ $$\angle P = 30^\circ$$ 4. **Find $\angle R$:** $$\angle R = 2 \times 30^\circ = 60^\circ$$ 5. **Summary of angles:** - $\angle P = 30^\circ$ - $\angle Q = 90^\circ$ - $\angle R = 60^\circ$ 6. **Use the Pythagorean theorem:** Since $\angle Q = 90^\circ$, side $PR$ is the hypotenuse, and sides $PQ$ and $QR$ are legs. The theorem states: $$PR^2 = PQ^2 + QR^2$$ 7. **Additional relation given:** $$PR^2 + DQ^2 = DR^2 + PQ^2$$ This appears to be a relation involving points D, Q, and R, which is not fully defined here. Without more information about points D, Q, and R, we cannot proceed further with this equation. **Final answer:** The angles of triangle PQR are $30^\circ$, $60^\circ$, and $90^\circ$. The Pythagorean theorem applies as: $$PR^2 = PQ^2 + QR^2$$