1. **State the problem:** We are given a circle with points B, E, and A, and an angle at point A, $\angle BAC = 74^\circ$. We need to find the measure of $\angle BEA$. 2. **Identify the relevant theorem:** In a circle, the measure of an inscribed angle is half the measure of its intercepted arc. 3. **Analyze the given angle:** $\angle BAC = 74^\circ$ is an angle formed by the chord AB and the tangent line at A (since the horizontal line through A is tangent). 4. **Use the tangent-secant angle theorem:** The angle between a tangent and a chord through the point of tangency equals half the measure of the intercepted arc. 5. **Apply the theorem:** Since $\angle BAC = 74^\circ$, the intercepted arc BE has measure $2 \times 74^\circ = 148^\circ$. 6. **Find $\angle BEA$:** $\angle BEA$ is an inscribed angle that intercepts the same arc BE, so its measure is half the arc measure. $$\angle BEA = \frac{148^\circ}{2} = 74^\circ$$ 7. **Final answer:** $$\boxed{74^\circ}$$