To solve for \( m\angle D \) in quadrilateral \( ABCD \) which is inscribed in circle \( O \), we know that the opposite angles of a cyclic quadrilateral sum to \( 180^\circ \). That is:
\[ m\angle B + m\angle D = 180^\circ \]
Given:
\[ m\angle B = 4y + 27 \] \[ m\angle D = y - 12 \]
We can set up the equation:
\[ (4y + 27) + (y - 12) = 180 \]
Now, simplify the equation:
\[ 4y + 27 + y - 12 = 180 \] \[ 5y + 15 = 180 \]
Next, subtract \( 15 \) from both sides:
\[ 5y = 165 \]
Now, divide by \( 5 \):
\[ y = 33 \]
Now that we have found \( y \), we can find \( m\angle D \):
\[ m\angle D = y - 12 = 33 - 12 = 21 \]
Thus, the measure of angle \( D \) is:
\[ \boxed{21} \]
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