To find \( m \angle VUM \), we can use the fact that \(\overline{QS}\) and \(\overline{TV}\) are parallel lines. Given that \( m \angle VUR = 127^\circ \), we can use the properties of angles formed by a transversal intersecting parallel lines.
Assuming that angles \( VUR \) and \( VUM \) are related as corresponding angles, alternate interior angles, or same-side interior angles, we can identify their relationship based on their positions:
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If \( VUR \) and \( VUM \) are corresponding angles or alternate interior angles, then they are equal: \[ m \angle VUM = m \angle VUR = 127^\circ \]
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If \( VUR \) and \( VUM \) are same-side interior angles, their measures will add up to \( 180^\circ \): \[ m \angle VUM = 180^\circ - m \angle VUR = 180^\circ - 127^\circ = 53^\circ \]
To determine which case applies, we need more information about the diagram or the relative position of angles \( VUR \) and \( VUM \).
For now, based on the information provided, if angles \( VUR \) and \( VUM \) are either corresponding or alternate interior angles, \( m \angle VUM = 127^\circ\). Otherwise, if they are same-side interior angles, \( m \angle VUM = 53^\circ\).
Please clarify if specific details of the angles or their positions are available for a more accurate answer!