1. **State the problem:** Given that angle A is circumscribed about circle O, find the measure of \(\angle A\) given that the central angle \(\angle BOC = 92^\circ\). 2. **Recall the property:** A circumscribed angle (tangent-secant angle) is equal to half the measure of the intercepted arc. Here, \(\angle A\) intercepts the arc \(\overset{\frown}{BC}\) subtended by central angle \(\angle BOC\). 3. **Apply the property:** Since \(\angle BOC = 92^\circ\) is the central angle, the intercepted arc \(\overset{\frown}{BC}\) also measures \(92^\circ\). 4. **Calculate \(\angle A\):** \n$$ \angle A = \frac{1}{2} \times 92^\circ = 46^\circ $$ 5. **Interpretation:** Therefore, the measure of \(\angle A\) is \(46^\circ\).