If angles \( j \) and \( k \) are supplementary, it means that their measures add up to \( 180^\circ \).
Given that \( m \angle j = 34^\circ \), we can find \( m \angle k \) using the following equation:
\[ m \angle j + m \angle k = 180^\circ \]
Substituting the given value for \( m \angle j \):
\[ 34^\circ + m \angle k = 180^\circ \]
To find \( m \angle k \), subtract \( 34^\circ \) from \( 180^\circ \):
\[ m \angle k = 180^\circ - 34^\circ = 146^\circ \]
Thus, \( m \angle k = 146^\circ \).