1. **State the problem:** We need to find the measure of angle $m\angle 1$ in a triangle where the other two angles are $58^\circ$ and $84^\circ$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always $180^\circ$. 3. **Set up the equation:** Let $m\angle 1 = x$. Then, $$58^\circ + 84^\circ + x = 180^\circ$$ 4. **Calculate the sum of the known angles:** $$58 + 84 = 142$$ 5. **Solve for $x$:** $$x = 180 - 142 = 38$$ 6. **Conclusion:** The measure of angle $m\angle 1$ is $38^\circ$.