1. **State the problem:** We have a triangle DEF with angles labeled as follows: angle E is 64°, angle D is 2x, and there is an adjacent angle of 42° at vertex F on the line DG. We need to find the value of $x$. 2. **Recall the angle sum property of a triangle:** The sum of the interior angles of any triangle is always 180°. That is, $$\text{angle } D + \text{angle } E + \text{angle } F = 180^\circ$$ 3. **Identify the angles inside triangle DEF:** We know angle E = 64°, angle D = 2x, and angle F is adjacent to the 42° angle on the straight line DG. Since DG is a straight line, the angles at F on the line sum to 180°. 4. **Calculate angle F inside the triangle:** The angle adjacent to angle F on the line is 42°, so $$\text{angle } F = 180^\circ - 42^\circ = 138^\circ$$ 5. **Set up the equation using the angle sum property:** $$2x + 64^\circ + 138^\circ = 180^\circ$$ 6. **Simplify the equation:** $$2x + 202^\circ = 180^\circ$$ 7. **Isolate $2x$:** $$2x = 180^\circ - 202^\circ$$ $$2x = -22^\circ$$ 8. **Solve for $x$:** $$x = \frac{-22^\circ}{2}$$ $$x = -11^\circ$$ 9. **Interpretation:** The value of $x$ is $-11$, which is not possible for an angle measure in this context, indicating a possible error in the problem setup or interpretation of the angles. **Final answer:** $$x = -11$$