1. **Problem statement:** We have a large triangle PQR with point S on segment PR, forming two smaller isosceles triangles PQS and SQR. The angle at vertex P in triangle PQR is $x^\circ$, and the angle at vertex Q in triangle SQR is also $x^\circ$. We want to analyze the relationship between these angles. 2. **Key properties:** In isosceles triangles, the angles opposite the equal sides are equal. 3. **Step 1:** Since triangle PQS is isosceles, the two sides marked equal imply the base angles are equal. Let the equal angles at Q and S in triangle PQS be $a$. 4. **Step 2:** Similarly, triangle SQR is isosceles with equal sides marked, so the base angles at S and R are equal. Let these angles be $b$. 5. **Step 3:** In triangle PQR, angle at P is $x$, and angle at Q is composed of angles from triangles PQS and SQR. Given the angle at Q in triangle SQR is $x$, we use this to relate the angles. 6. **Step 4:** Using the triangle angle sum property, sum of angles in triangle PQR is $180^\circ$: $$x + \angle Q + \angle R = 180^\circ$$ 7. **Step 5:** Using the isosceles properties and given equal angles, we find that the angle at P equals the angle at Q in triangle SQR, both equal to $x^\circ$. **Final conclusion:** The angle $x$ at vertex P in triangle PQR is equal to the angle $x$ at vertex Q in triangle SQR, consistent with the isosceles triangle properties and the given markings.