In a cyclic quadrilateral, the opposite angles are supplementary. This means that the sum of the measures of angle \( B \) and angle \( D \) is equal to \( 180^\circ \).
Given:
- \( m\angle B = 4y + 27 \)
- \( m\angle D = y - 12 \)
Using the property of supplementary angles, we write the equation: \[ m\angle B + m\angle D = 180^\circ \] Substituting the expressions for \( m\angle B \) and \( m\angle D \): \[ (4y + 27) + (y - 12) = 180 \]
Now, combine like terms: \[ 4y + y + 27 - 12 = 180 \] \[ 5y + 15 = 180 \]
Subtract \( 15 \) from both sides: \[ 5y = 165 \]
Now, divide by \( 5 \): \[ y = 33 \]
Next, we can find \( m\angle D \): \[ m\angle D = y - 12 = 33 - 12 = 21 \]
Thus, the measure of angle \( D \) is: \[ \boxed{21} \text{°} \]