1. **Stating the problem:** We have a circle with center O and points A, B, C, D, E on the circumference. Given that the sum of the inscribed angles $\angle BDA + \angle BCA + \angle BEA = 180^\circ$, we need to find the measure of the central angle $\angle AOB$. 2. **Recall the key property:** An inscribed angle in a circle is half the measure of the central angle that subtends the same arc. That is, if an inscribed angle intercepts an arc, the central angle intercepting the same arc is twice the inscribed angle. 3. **Analyze the given angles:** Each of the angles $\angle BDA$, $\angle BCA$, and $\angle BEA$ are inscribed angles subtending arcs on the circle. 4. **Sum of inscribed angles:** Given: $$\angle BDA + \angle BCA + \angle BEA = 180^\circ$$ 5. **Relate inscribed angles to arcs:** Each inscribed angle equals half the measure of its intercepted arc. Let the arcs intercepted by these angles be $m(BA)$, $m(CA)$, and $m(EA)$ respectively. Then: $$\angle BDA = \frac{1}{2} m(BA), \quad \angle BCA = \frac{1}{2} m(CA), \quad \angle BEA = \frac{1}{2} m(EA)$$ 6. **Sum of arcs:** Sum of inscribed angles is: $$\frac{1}{2} [m(BA) + m(CA) + m(EA)] = 180^\circ$$ Multiply both sides by 2: $$m(BA) + m(CA) + m(EA) = 360^\circ$$ 7. **Interpretation:** The sum of these arcs is the entire circle (360°). Since arcs $BA$, $CA$, and $EA$ together cover the whole circle, the arcs between points B, C, D, E, and A sum to 360°. 8. **Find $\angle AOB$:** The central angle $\angle AOB$ intercepts arc $AB$. Since the arcs $BA$, $CA$, and $EA$ cover the entire circle, the arc $AB$ is the entire circle minus these arcs, which is 0°. But since $BA$ is the same as $AB$ (arc direction matters), the central angle $\angle AOB$ corresponds to the sum of these arcs. Therefore, the central angle $\angle AOB$ is: $$\angle AOB = 360^\circ$$ But a central angle cannot be 360°, so the problem implies that $\angle AOB$ is the sum of the inscribed angles multiplied by 2: Since the sum of inscribed angles is 180°, the central angle $\angle AOB$ is: $$\angle AOB = 2 \times 180^\circ = 360^\circ$$ This means $\angle AOB$ is a straight angle, or effectively the full circle. **Final answer:** $$\boxed{180^\circ}$$ Because the central angle corresponding to the sum of inscribed angles 180° is 180°, the measure of $\angle AOB$ is $180^\circ$.