1. **State the problem:** We need to find the measure of angle 1, denoted as $m \angle 1$, in a triangle where two other angles are given as $42^\circ$ and $35^\circ$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always $180^\circ$. This means: $$m \angle 1 + 42^\circ + 35^\circ = 180^\circ$$ 3. **Set up the equation:** $$m \angle 1 = 180^\circ - 42^\circ - 35^\circ$$ 4. **Calculate:** $$m \angle 1 = 180^\circ - 77^\circ = 103^\circ$$ 5. **Conclusion:** The measure of angle 1 is $103^\circ$.