1. **Stating the problem:** We are given a circle with center O and points A, B, C, D, E on the circumference. We know three arc angles near point E: 12°, 14°, and 73°, and we need to find the angle $\theta$ near point B inside the circle. 2. **Relevant formula:** The angle formed inside a circle by two chords intersecting at a point inside the circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Mathematically, if two chords intersect inside the circle creating angle $\theta$, then: $$\theta = \frac{1}{2}(\text{arc}_1 + \text{arc}_2)$$ where $\text{arc}_1$ and $\text{arc}_2$ are the measures of the arcs intercepted by the angle and its vertical angle. 3. **Identify arcs intercepted by angle $\theta$:** Since $\theta$ is near point B, the intercepted arcs are the arcs opposite to angle $\theta$ formed by chords through B. From the problem, the arcs near E are 12°, 14°, and 73°, so the arcs intercepted by $\theta$ are the arcs $AE = 12^\circ$, $BE = 14^\circ$, and $DE = 73^\circ$. 4. **Calculate the sum of intercepted arcs:** The two arcs intercepted by angle $\theta$ and its vertical angle are $12^\circ + 73^\circ = 85^\circ$. 5. **Calculate $\theta$:** Using the formula: $$\theta = \frac{1}{2} \times 85^\circ = 42.5^\circ$$ 6. **Conclusion:** The value of angle $\theta$ is $42.5^\circ$.