To find the measure of angle \( DBC \), we start with the information given:
- \( m \angle ABC = 160^\circ \)
- \( m \angle ABD = 92^\circ \)
Since angles \( ABD \) and \( DBC \) are adjacent angles and lie within angle \( ABC \), the sum of the measures of angles \( ABD \) and \( DBC \) must equal the measure of angle \( ABC \):
\[ m \angle ABC = m \angle ABD + m \angle DBC \]
Substituting the known values:
\[ 160^\circ = 92^\circ + m \angle DBC \]
To find \( m \angle DBC \), we can rearrange the equation:
\[ m \angle DBC = 160^\circ - 92^\circ \] \[ m \angle DBC = 68^\circ \]
Therefore, the measure of angle \( DBC \) is \( \boxed{68} \).