1. **Stating the problem:** We are given a geometric figure with angles and arcs, and we need to find the value of angle $\alpha$. The figure includes a large semicircle with a triangle inscribed, where the left base angle is $\alpha$, and the right base angle is $64^\circ$. The triangle's top vertex lies on the semicircle. 2. **Key geometric fact:** In a semicircle, any triangle inscribed with the diameter as one side is a right triangle. The angle opposite the diameter is $90^\circ$. 3. **Using the triangle angle sum rule:** The sum of angles in any triangle is $180^\circ$. Let the three angles be $\alpha$, $64^\circ$, and the right angle $90^\circ$. 4. **Calculate $\alpha$:** $$\alpha + 64^\circ + 90^\circ = 180^\circ$$ $$\alpha + 154^\circ = 180^\circ$$ $$\alpha = 180^\circ - 154^\circ$$ $$\alpha = 26^\circ$$ 5. **Conclusion:** The value of $\alpha$ is $26^\circ$.