1. **State the problem:** We are given a triangle DEF with angles at vertices D, E, and F labeled as $x$, $2x$, and $5x$ respectively. We need to find the measure of angle $D$, which is $m\angle D = x$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $180^\circ$. Therefore, we have: $$x + 2x + 5x = 180$$ 3. **Combine like terms:** $$8x = 180$$ 4. **Solve for $x$:** $$x = \frac{180}{8}$$ 5. **Simplify the fraction:** $$x = \frac{\cancel{180}^{\div 4}}{\cancel{8}^{\div 4}} = \frac{45}{2} = 22.5$$ 6. **Conclusion:** The measure of angle $D$ is $m\angle D = x = 22.5^\circ$.