Supplementary angles are two angles that add up to 180 degrees. Given that angle \( \angle ABC \) measures \( 135^\circ \) and angle \( \angle CBD \) measures \( 9x^\circ \), we can set up the equation based on the definition of supplementary angles:
\[ m\angle ABC + m\angle CBD = 180^\circ \]
Substituting the given angle measures into the equation, we have:
\[ 135^\circ + 9x^\circ = 180^\circ \]
To solve for \( x \), we can rearrange the equation:
\[ 9x^\circ = 180^\circ - 135^\circ \]
Calculating the right side:
\[ 9x^\circ = 45^\circ \]
Finally, divide both sides by 9 to find \( x \):
\[ x = \frac{45^\circ}{9} \] \[ x = 5 \]
Thus, the equation to solve for \( x \) is:
\[ 135 + 9x = 180 \]