1. The problem asks to find the measure of angle $\angle 1$ in a triangle where two other angles are given as $76^\circ$ and $51^\circ$. 2. Recall the Triangle Angle Sum Theorem: The sum of the interior angles of a triangle is always $180^\circ$. 3. Using this theorem, we can write the equation: $$\angle 1 + 76^\circ + 51^\circ = 180^\circ$$ 4. Combine the known angles: $$76^\circ + 51^\circ = 127^\circ$$ 5. Substitute back into the equation: $$\angle 1 + 127^\circ = 180^\circ$$ 6. Solve for $\angle 1$ by subtracting $127^\circ$ from both sides: $$\angle 1 = 180^\circ - 127^\circ = 53^\circ$$ 7. Therefore, the measure of $\angle 1$ is $53^\circ$.