In a rhombus, opposite angles are equal and the sum of the angles in any quadrilateral is 360°. Since rhombus ABCD has \( m\angle A = 114^\circ \), it follows that \( m\angle C = 114^\circ \) as well (since \( \angle A \) and \( \angle C \) are opposite angles).
Now, to find the measures of angles B and D, we can use the fact that the sum of all angles in a quadrilateral is \( 360^\circ \):
\[ m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ \]
Substituting the known values:
\[ 114^\circ + m\angle B + 114^\circ + m\angle D = 360^\circ \]
This simplifies to:
\[ 228^\circ + m\angle B + m\angle D = 360^\circ \]
Now, simplify further to find:
\[ m\angle B + m\angle D = 360^\circ - 228^\circ \] \[ m\angle B + m\angle D = 132^\circ \]
Since \( m\angle B \) and \( m\angle D \) are also equal (opposite angles in a rhombus):
\[ m\angle B = m\angle D \]
We can substitute \( m\angle D \) with \( m\angle B \):
\[ m\angle B + m\angle B = 132^\circ \] \[ 2m\angle B = 132^\circ \]
Dividing both sides by 2:
\[ m\angle B = 66^\circ \]
Thus, the measure of angle B in rhombus ABCD is \( \boxed{66^\circ} \).
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