To find the measure of angle \( \angle JHI \), we can use the fact that the angles \( \angle GHJ \) and \( \angle JHI \) are adjacent angles that together form \( \angle GHI \).
We know:
- Measure of \( \angle GHI = 135^\circ \)
- Measure of \( \angle GHJ = 112^\circ \)
The relationship can be described as:
\[ \angle GHI = \angle GHJ + \angle JHI \]
Substituting the known values into the equation:
\[ 135^\circ = 112^\circ + \angle JHI \]
To find \( \angle JHI \), we subtract \( 112^\circ \) from \( 135^\circ \):
\[ \angle JHI = 135^\circ - 112^\circ = 23^\circ \]
Therefore, the measure of \( \angle JHI \) is \( \boxed{23^\circ} \).