1. **Problem statement:** Given a circle with diameter AB and a point C on the circumference, angle ABC is 38°. We need to find the measure of angle ACB. 2. **Key property:** In a circle, an angle inscribed in a semicircle (where the side is a diameter) is a right angle (90°). This means angle ACB is 90° because AB is the diameter. 3. **Triangle ABC:** Since AB is the diameter, triangle ABC is a right triangle with the right angle at C. 4. **Sum of angles in triangle:** The sum of angles in triangle ABC is 180°. So, $$\angle ABC + \angle BAC + \angle ACB = 180^\circ$$ 5. **Substitute known values:** We know $\angle ABC = 38^\circ$ and $\angle ACB = 90^\circ$, so $$38^\circ + \angle BAC + 90^\circ = 180^\circ$$ 6. **Solve for $\angle BAC$:** $$\angle BAC = 180^\circ - 38^\circ - 90^\circ = 52^\circ$$ **Final answer:** $$\boxed{52^\circ}$$ is the measure of angle BAC.