1. **Stating the problem:** We are given a circle with points A, C, D, E, and an angle $\angle CAE = 24^\circ$ and an arc $\overset{\frown}{DE} = 83^\circ$. We need to find the measure of angle $\hat{X}$ (assumed to be $\angle X$ related to these points). 2. **Understanding the problem:** In circle geometry, angles formed by chords and arcs have specific relationships. Here, $\angle CAE$ is an angle formed by points on the circle, and $\overset{\frown}{DE}$ is the measure of the arc between points D and E. 3. **Key formula:** The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 4. Since $\angle CAE = 24^\circ$ and arc $\overset{\frown}{DE} = 83^\circ$, we use the property that the angle formed by two chords intersecting inside the circle is: $$\angle = \frac{1}{2} (\text{arc}_1 + \text{arc}_2)$$ where $\text{arc}_1$ and $\text{arc}_2$ are the arcs intercepted by the angle and its vertical angle. 5. We want to find $\hat{X}$, which is likely the angle formed by the secant lines or chords related to points D and E. 6. Using the given $\angle CAE = 24^\circ$, and knowing that the arc $\overset{\frown}{DE} = 83^\circ$, we can find the other arc intercepted by $\angle CAE$ using the circle's total 360°: $$\text{arc } CE = 2 \times \angle CAE = 2 \times 24^\circ = 48^\circ$$ 7. Since $\angle CAE$ intercepts arcs $CE$ and $DE$, the sum of these arcs is: $$48^\circ + 83^\circ = 131^\circ$$ 8. The angle $\hat{X}$ formed by the chords or secants related to these arcs is half the difference of the arcs intercepted by the secants: $$\hat{X} = \frac{1}{2} |\overset{\frown}{DE} - \overset{\frown}{CE}| = \frac{1}{2} |83^\circ - 48^\circ| = \frac{1}{2} \times 35^\circ = 17.5^\circ$$ **Final answer:** $$\boxed{17.5^\circ}$$