1. **Problem Statement:** We are given a circle with a tangent line $\overrightarrow{GH}$ at point $H$ and a chord $\overline{FH}$. The angle between the tangent $\overrightarrow{GH}$ and the chord $\overline{FH}$ is $44^\circ$. The measure of the arc $\overset{\frown}{EF}$ is $133^\circ$. We need to find the measure of the minor arc $\overset{\frown}{FH}$. 2. **Formula Used:** The angle formed between a tangent and a chord is half the measure of the intercepted arc. Mathematically, $$\text{Angle} = \frac{1}{2} \times m\overset{\frown}{FH}$$ where $m\overset{\frown}{FH}$ is the measure of the arc intercepted by the angle. 3. **Important Rule:** The tangent-chord angle equals half the measure of the intercepted arc. 4. **Given:** - Angle between tangent and chord $= 44^\circ$ - Arc $\overset{\frown}{EF} = 133^\circ$ 5. **Find:** $m\overset{\frown}{FH}$ 6. **Step-by-step Solution:** - The angle between tangent $\overrightarrow{GH}$ and chord $\overline{FH}$ intercepts arc $\overset{\frown}{FH}$. - Using the formula: $$44 = \frac{1}{2} \times m\overset{\frown}{FH}$$ - Multiply both sides by 2: $$2 \times 44 = \cancel{2} \times \frac{1}{\cancel{2}} \times m\overset{\frown}{FH}$$ $$88 = m\overset{\frown}{FH}$$ 7. **Answer:** $$m\overset{\frown}{FH} = 88^\circ$$ This means the minor arc $\overset{\frown}{FH}$ measures $88^\circ$.