The lines a and b are parallel and the transversal line c intersects these parallel lines.
The angles in the matching corners are called "corresponding angles" and they are equal.
This means that the following angles are equal
∠1 = ∠10
∠9 = ∠14
∠8 = ∠13
∠2 = ∠11 + ∠12
Since we are given that ∠10 = 77°, then ∠1 must also be 77°
Therefore, the measure of the missing angle ∠1 is 77°