To determine \( m\angle ADU \) in an inscribed quadrilateral \( QUAD \), we can use the property that opposite angles of a cyclic quadrilateral (a quadrilateral inscribed in a circle) are supplementary. This means that the sum of the measures of opposite angles is equal to 180 degrees.
We know the following angles:
- \( m\angle QUA = 55^\circ \)
- \( m\angle AUQ = 112^\circ \)
- \( m\angle UAD = 100^\circ \)
Since \( QUAD \) is a cyclic quadrilateral, we can apply the following relationships:
- \( m\angle UAD + m\angle QUA = 180^\circ \)
- \( m\angle AUQ + m\angle ADU = 180^\circ \)
From the first equation: \[ m\angle UAD + m\angle QUA = 100^\circ + 55^\circ = 155^\circ \] Since \( m\angle UAD + m\angle QUA \) does not equal \( 180^\circ \), we confirm that these angles are not opposite but adjacent. We need to check the opposite angle \( ADU \).
To find \( m\angle ADU \): Using the second equation: \[ m\angle AUQ + m\angle ADU = 180^\circ \] \[ 112^\circ + m\angle ADU = 180^\circ \] Now, solving for \( m\angle ADU \): \[ m\angle ADU = 180^\circ - 112^\circ = 68^\circ \]
Thus, the measure \( m\angle ADU \) is: \[ \boxed{68^\circ} \]
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