1. The problem asks to find the measure of angle $\angle 1$ in a triangle where the other two angles are $84^\circ$ and $66^\circ$. 2. Recall the rule that the sum of the interior angles in any triangle is always $180^\circ$. 3. Use the formula: $$m\angle 1 + 84^\circ + 66^\circ = 180^\circ$$ 4. Combine the known angles: $$84^\circ + 66^\circ = 150^\circ$$ 5. Substitute back: $$m\angle 1 + 150^\circ = 180^\circ$$ 6. Solve for $m\angle 1$: $$m\angle 1 = 180^\circ - 150^\circ = 30^\circ$$ 7. Therefore, the measure of angle $\angle 1$ is $30^\circ$.