To determine the measure of angle \( A \) in the inscribed quadrilateral where the measure of angle \( AMB \) is given as \( 54^\circ \), we need to use the property of inscribed angles.
In an inscribed quadrilateral, the opposite angles are supplementary. This means that:
\[ m\angle A + m\angle C = 180^\circ \] \[ m\angle B + m\angle D = 180^\circ \]
The angle \( AMB \) is an inscribed angle that intercepts the arc opposite it, which is related to angle \( A \) and angle \( B \) in this quadrilateral.
By the inscribed angle theorem, \( m\angle AMB \) is equal to half the measure of the arc that it intercepts. Therefore, if we know \( m\angle AMB = 54^\circ \), it means that:
\[ m\angle A + m\angle B = 54^\circ \text{ (Because \( A \) and \( B \) share the same intercepted arc with } AMB) \]
Now, if you do not have additional data about the other angles, we cannot determine the exact values of \( m\angle A \) and \( m\angle B \) individually. However, if you're only looking for \( m\angle A \), it can be represented in relation to \( m\angle B \) as follows:
\[ m\angle A + m\angle B = 54^\circ \]
From the given information, we cannot isolate \( m\angle A \) without more information about \( m\angle B \).
If you have additional angles in the quadrilateral or specific relationships between the angles, please provide that information to further assist in calculating \( m\angle A \). If angle \( B \) or any other angle is known, we could find \( A \) using the above equations.
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