To find the measure of angle \( \angle U \), we can use the information given. From the problem, we have the following angles:
- \( m\angle A = 93^\circ \)
- \( mAU \angle Q = 126^\circ \)
- \( mQU \angle Q = 90^\circ \)
We can summarize these angles into a triangle involving points A, U, and Q. Let's write down the specific angles that form at point U:
In triangle AUQ, the sum of the angles must equal \( 180^\circ \).
The angles can be expressed as:
- \( m\angle A = 93^\circ \)
- \( mAUQ = 126^\circ \)
- \( mQUQ = 90^\circ \)
Now, we know that: \[ m\angle A + mAUQ + mQUQ + m\angle U = 180^\circ \]
However, we realize there is a mistake because \( mAUQ \) and \( mQUQ \) are not the internal angles at U in triangle AUQ. Instead, we should look closer, particularly since \( mQUQ = 90^\circ \) appears to be more of a right angle.
In any triangle: \[ m\angle U = 180^\circ - (m\angle A + mAUQ) \]
Insert the known measures into the equation: \[ m\angle U = 180^\circ - (93^\circ + 126^\circ) \] \[ = 180^\circ - 219^\circ \] However, that part leads to a contradiction since angles in triangle shouldn't equal more than \( 180^\circ \).
Thus, backtrack:
Since \( \angle AUQ \) appears to be a corresponding angle:
- We should re-assert that angle at U taking the rest of angle space. We should carefully check angles relative to AU and QU where angles overlap and request measure only left for \( \angle U \) under remaining triangle measures closer.
Thus each situation defines two angles. Therefore we can clarify:
- Correct spaces yielding a triangle limited to understand left over oversight of counter directions:
This still simplifies to review: Right angle triangle consideration where \( 90 + mU + 93 + 126\) verifies into possible circle complete.
With that klare:
\[ m\angle U= 360 - 90 - 93 - 126= 51^\circ \]
Thus, the correct measure is: \[ \boxed{51^\circ} \]