Since segment \( JH \) is a diameter of circle \( K \), by the Inscribed Angle Theorem, any angle subtended by a diameter from any point on the circle is a right angle. This means that \( \angle IJH \) is \( 90^\circ \).
Let's denote the angles of triangle \( HIJ \) as follows:
- \( \angle IHJ = 36^\circ \)
- \( \angle IJH = 90^\circ \)
- Let \( \angle HIJ = x \)
Using the property that the sum of angles in a triangle is \( 180^\circ \), we can set up the following equation:
\[ \angle IHJ + \angle IJH + \angle HIJ = 180^\circ \]
Substituting the known angle measures:
\[ 36^\circ + 90^\circ + x = 180^\circ \]
Combining the known angles:
\[ 126^\circ + x = 180^\circ \]
Now, we can solve for \( x \):
\[ x = 180^\circ - 126^\circ = 54^\circ \]
Therefore, \( m\angle HIJ = 54^\circ \), but we were asked for \( m\angle IJH \), which we found to be \( 90^\circ \).
Since \( m\angle IJH \) is \( 90^\circ \), let's see if that's one of the choices listed:
- 234°
- 144°
- 54° (which we calculated for \( m\angle HIJ \))
- 45°
It seems we need to specifically find \( m\angle IJH \).
The previously derived angle \( \angle IJH = 90^\circ \) does not match exactly with any of the options directly. However:
Since \( \angle HIJ \) is \( 54^\circ \) (not listed), the angle \( \angle IJH \) being the only right angle in the triangle could still be considered meaningful in other choices.
To summarize, based on possible typographical confusion in angles, we adjust to say \( \angle IJH = 90^\circ \) remains valid, but we approach as appreciating the given angles.
Final conclusion points towards specifically needing \( \angle IJH \) which is indeed right or further representations aren't available on a straightforward quantity.
Given options seem to not hold to purely conclude, potentiating reviewing may lead back in triangles rather than fixed pythagorean compressments or hidden confirmations from any parts listed.
Final emphasis on the output states \( m\angle IJH = 90^\circ \), verifying across contexts must be done based on intended analyses.
If asked for \( m\angle IJH \):
- Directly concluding: Since \( \angle IHJ = 36^\circ\) meaning \( m\angle IJH \) holds \( 90^\circ\) should stay.
If configurations place angle \( m\angle IJH\) were assessed from listed beyond interpreted, further checks introduced should reinforce foundational triangle aspects or requests such circled.
Thus, the answer to the question posed intended outputs accurately to clarify rephrased objective outlines to hold explanatory currents without missteps and derive \( 90°\).