1. **Problem statement:** Given a circle with two tangent lines $ \overrightarrow{EF} $ and $ \overrightarrow{ED} $ touching the circle at points F and D respectively, and the angle inside the circle between points D and F is $101^\circ$. We need to find the measure of the angle $ \angle DEF $ formed outside the circle by the two tangents. 2. **Relevant formula:** The angle formed outside the circle by two tangents is half the difference of the measures of the intercepted arcs. Since the two tangents touch the circle at points D and F, the angle $ \angle DEF $ is given by: $$\angle DEF = \frac{1}{2} \times \text{difference of intercepted arcs}$$ 3. **Important rule:** The two tangents from a point outside a circle are equal in length and the angle between them is related to the arcs they intercept. 4. **Given:** The angle inside the circle between points D and F is $101^\circ$. This is the measure of the arc between D and F. 5. **Calculate the other arc:** The circle's total degrees is $360^\circ$, so the other arc intercepted by the tangents is: $$360^\circ - 101^\circ = 259^\circ$$ 6. **Calculate $ \angle DEF $:** $$\angle DEF = \frac{1}{2} \times (259^\circ - 101^\circ) = \frac{1}{2} \times 158^\circ = 79^\circ$$ 7. **Answer:** The measure of $ \angle DEF $ is $79^\circ$. This means the angle formed outside the circle by the two tangents is 79 degrees.