Supplementary angles are two angles that add up to 180 degrees. In this problem, we are given that:
- \( m \angle ABC = 15^\circ \)
- \( m \angle CBD = x \)
Since these two angles are supplementary, we can write the equation:
\[ m \angle ABC + m \angle CBD = 180^\circ \]
Substituting the known value:
\[ 15^\circ + x = 180^\circ \]
To find \( x \), we can solve for it:
\[ x = 180^\circ - 15^\circ \]
\[ x = 165^\circ \]
Therefore, the value of \( m \angle CBD \) is 165 degrees.