When two lines intersect, they create pairs of vertical angles, which are equal, and adjacent angles that are supplementary (add up to 180 degrees).
Given that \( m\angle C = 75^\circ \), we know that the angle adjacent to angle C, which is angle B, must be:
\[ m\angle B + m\angle C = 180^\circ \]
Substituting \( m\angle C \):
\[ m\angle B + 75^\circ = 180^\circ \]
Solving for \( m\angle B \):
\[ m\angle B = 180^\circ - 75^\circ = 105^\circ \]
Now, angle A is adjacent to angle B, so we apply the same supplementary relationship:
\[ m\angle A + m\angle B = 180^\circ \]
Substituting \( m\angle B \):
\[ m\angle A + 105^\circ = 180^\circ \]
Solving for \( m\angle A \):
\[ m\angle A = 180^\circ - 105^\circ = 75^\circ \]
Thus, the measure of angle A, \( m\angle A \), is:
\[ \boxed{75} \]