To find the measure of angle \( m \angle BCA \), we can use the properties of parallel lines and transversals.
Given that \( BD \) and \( EG \) are parallel lines, and \( EFH \) is a transversal that intersects these lines, we can deduce the relationships between the angles formed.
Since \( m \angle EFH = 127^\circ \), we know that \( \angle EFH \) and \( \angle BCA \) are corresponding angles. Corresponding angles are equal when a transversal crosses parallel lines.
Therefore, we have:
\[ m \angle BCA = m \angle EFH = 127^\circ \]
Thus, the measure of angle \( BCA \) is \( 127^\circ \).