To find the measure of angle \( JHI \), we can use the fact that the sum of the measures of adjacent angles should equal the measure of the obtuse angle \( GHI \).
Given:
- The measure of angle \( GHI = 135^\circ \)
- The measure of angle \( GHJ = 112^\circ \)
Since angles \( GHJ \) and \( JHI \) are adjacent angles that sum up to angle \( GHI \), we can write the equation:
\[ GHJ + JHI = GHI \]
Substituting the known values:
\[ 112^\circ + JHI = 135^\circ \]
Now, solve for \( JHI \):
\[ JHI = 135^\circ - 112^\circ \] \[ JHI = 23^\circ \]
Therefore, the measure of angle \( JHI \) is \( \boxed{23} \).