1. The problem asks for the measure of angle $\angle 1$ in a triangle where the other two angles are $34^\circ$ and $116^\circ$. 2. Recall the Triangle Angle Sum Theorem: The sum of the interior angles of a triangle is always $180^\circ$. 3. Using the theorem, we write the equation: $$34^\circ + 116^\circ + m\angle 1 = 180^\circ$$ 4. Add the known angles: $$150^\circ + m\angle 1 = 180^\circ$$ 5. Subtract $150^\circ$ from both sides to isolate $m\angle 1$: $$m\angle 1 = 180^\circ - 150^\circ$$ 6. Calculate the difference: $$m\angle 1 = 30^\circ$$ Therefore, the measure of angle $\angle 1$ is $30^\circ$.