1. **Stating the problem:** We have a triangle inscribed in a circle with one angle inside the triangle labeled $45^\circ$, an exterior angle near the chord labeled $88^\circ$, and we want to find the unknown angle $x$ opposite the $45^\circ$ angle. 2. **Key rule:** The exterior angle of a triangle is equal to the sum of the two opposite interior angles. Here, the exterior angle is $88^\circ$. 3. **Set up the equation:** Let the three angles of the triangle be $45^\circ$, $x$, and the third angle $y$. The exterior angle $88^\circ$ equals the sum of the two opposite interior angles: $$88^\circ = 45^\circ + y$$ 4. **Solve for $y$:** $$y = 88^\circ - 45^\circ = 43^\circ$$ 5. **Sum of angles in a triangle:** $$45^\circ + x + y = 180^\circ$$ Substitute $y = 43^\circ$: $$45^\circ + x + 43^\circ = 180^\circ$$ 6. **Simplify and solve for $x$:** $$x + 88^\circ = 180^\circ$$ $$x = 180^\circ - 88^\circ = 92^\circ$$ **Final answer:** $$\boxed{92^\circ}$$