1. **State the problem:** We need to find the measure of angle $m\angle 1$ in a triangle where the other two angles are $32^\circ$ and $133^\circ$. 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, $$32^\circ + 133^\circ + x = 180^\circ$$ 4. **Calculate the sum of the known angles:** $$32 + 133 = 165$$ 5. **Solve for $x$:** $$x = 180 - 165 = 15$$ 6. **Conclusion:** The measure of angle $m\angle 1$ is $15^\circ$.