In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the opposite angles are supplementary. This means that the sum of angles \( \angle B \) and \( \angle D \) is \( 180^\circ \).
Given: \[ m\angle B = 4y + 27 \] \[ m\angle D = y - 12 \]
According to the property of cyclic quadrilaterals: \[ m\angle B + m\angle D = 180^\circ \]
Substituting the expressions for \( m\angle B \) and \( m\angle D \) into this equation, we get: \[ (4y + 27) + (y - 12) = 180 \]
Now, simplify the equation: \[ 4y + 27 + y - 12 = 180 \] \[ 5y + 15 = 180 \]
Next, isolate \( y \) by subtracting 15 from both sides: \[ 5y = 180 - 15 \] \[ 5y = 165 \]
Now, divide both sides by 5: \[ y = \frac{165}{5} = 33 \]
Now that we have the value of \( y \), we can find \( m\angle D \): \[ m\angle D = y - 12 = 33 - 12 = 21 \]
Thus, the measure of angle \( D \) is: \[ \boxed{21^\circ} \]
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