1. **Problem Statement:** Given a circle with diameter $\overline{EF}$ and tangent $\overline{EH}$ at point $E$, and the measure of arc $\widehat{EFG} = 204^\circ$, find: (a) $m \angle GEH$ (b) $m \angle FEG$ 2. **Key Concepts and Formulas:** - The diameter $\overline{EF}$ means $E$ and $F$ are endpoints of a diameter, so the arc $\widehat{EF}$ is a semicircle of $180^\circ$. - The tangent line $\overline{EH}$ at point $E$ is perpendicular to the radius $\overline{EF}$, so $\angle FEH = 90^\circ$. - The measure of an inscribed angle is half the measure of its intercepted arc. - The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. 3. **Find the measure of arc $\widehat{FG}$:** Since $\widehat{EFG} = 204^\circ$ and $\widehat{EF}$ is a semicircle of $180^\circ$, the arc $\widehat{FG} = 204^\circ - 180^\circ = 24^\circ$. 4. **Find $m \angle GEH$:** $\angle GEH$ is formed by the tangent $\overline{EH}$ and chord $\overline{EG}$. The measure of this angle is half the measure of the intercepted arc $\widehat{FG}$. Therefore, $$m \angle GEH = \frac{1}{2} \times 24^\circ = 12^\circ.$$ 5. **Find $m \angle FEG$:** $\angle FEG$ is an inscribed angle intercepting arc $\widehat{FG}$. The measure of an inscribed angle is half the measure of its intercepted arc. Therefore, $$m \angle FEG = \frac{1}{2} \times 24^\circ = 12^\circ.$$ **Final answers:** (a) $m \angle GEH = 12^\circ$ (b) $m \angle FEG = 12^\circ$