1. **Problem Statement:** We have two concentric circles with a smaller circle inside a larger one. A square is inscribed around the smaller circle, touching it at each corner. The angles at the corners of the square inside the smaller circle are marked as $\angle x$. Additionally, there are two angles marked $\angle x$ adjacent to tangent lines outside the circles. We need to find the values of the 5 unknown angles marked $\angle x$. 2. **Key Concepts:** - The square's corners touching the smaller circle means the circle is inscribed in the square. - Each corner angle of a square is $90^\circ$. - The angles marked $\angle x$ inside the smaller circle at the square's corners are formed by the tangent and radius lines. - Tangent to a circle is perpendicular to the radius at the point of tangency. 3. **Step-by-step solution:** - Since the square is inscribed around the smaller circle, the radius drawn to the point of tangency is perpendicular to the square's side. - At each corner of the square, the angle between the radius and the square's side is $90^\circ$. - The angles marked $\angle x$ inside the smaller circle at the corners are formed by the radius and the tangent line (which is the square's side), so each $\angle x = 90^\circ$. - Outside the circles, the angles marked $\angle x$ adjacent to the tangent lines are also formed between the tangent and the radius to the point of tangency, so these are also $90^\circ$. 4. **Conclusion:** All 5 unknown angles marked $\angle x$ are right angles, so: $$\boxed{\angle x = 90^\circ}$$