1. **Problem statement:** We are given a circle with center O, points A and B on the circumference, and a tangent line at point B. The angle between line OB and the tangent line PB is 78°. 2. **Key fact:** The angle between a radius and a tangent at the point of tangency is 90°. 3. **Given:** \(\angle PBO = 78^\circ\). 4. **Find:** The angle \(x = \angle AOB\) inside the circle. 5. Since \(OB\) is a radius and \(PB\) is tangent at B, \(\angle PBO = 78^\circ\) and \(\angle OBA = 90^\circ\) (radius-tangent angle). 6. The angle \(x = \angle AOB\) is the central angle subtended by arc AB. 7. The angle between the radius and tangent is 90°, so the angle between the tangent and the chord \(AB\) is \(90^\circ - 78^\circ = 12^\circ\). 8. By the tangent-chord angle theorem, the angle between the tangent and chord equals the angle in the alternate segment, which is \(x\). 9. Therefore, \(x = 12^\circ\). **Final answer:** \(x = 12^\circ\).