In a rhombus, opposite angles are equal, and the diagonals bisect each other at right angles. Given that \( m\angle A = 114^\circ \), we know that:
- \( m\angle C = m\angle A = 114^\circ \) (because opposite angles of a rhombus are equal).
The sum of the angles in any quadrilateral is \( 360^\circ \). Therefore, we can find \( m\angle B \) and \( m\angle D \) using the fact that:
\[ m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ \] Substituting the known values:
\[ 114^\circ + m\angle B + 114^\circ + m\angle D = 360^\circ \]
This simplifies to:
\[ 228^\circ + m\angle B + m\angle D = 360^\circ \]
Since \( m\angle B = m\angle D \) (opposite angles are equal), we can let \( m\angle B = x \):
\[ 228^\circ + x + x = 360^\circ \] \[ 228^\circ + 2x = 360^\circ \] \[ 2x = 360^\circ - 228^\circ \] \[ 2x = 132^\circ \] \[ x = 66^\circ \]
Thus, \( m\angle B = 66^\circ \).
The answer is \( \boxed{66^\circ} \).
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