\( X Z \) is the bisector of \( \angle W X Y \). If \( m \angle W X Y=9 x+12 \) and \( m \angle Y X Z=6 x-6 \), what is \( m \angle W X Z \) ?

To find \( m \angle W X Z \), we first use the fact that \( X Z \) is the bisector of \( \angle W X Y \). This means that: \[ m \angle W X Z = m \angle Y X Z \] From the problem, we know: - \( m \angle W X Y = 9x + 12 \) - \( m \angle Y X Z = 6x - 6 \) Since \( X Z \) is the bisector, we can set up the following equation: \[ m \angle W X Z = m \angle Y X Z = \frac{1}{2} m \angle W X Y \] So, we can write: \[ m \angle W X Y = m \angle W X Z + m \angle Y X Z \] Substituting the expressions we have gives: \[ 9x + 12 = (6x - 6) + (6x - 6) \] Simplifying this gives: \[ 9x + 12 = 6x - 6 + 6x - 6 \] Combine like terms: \[ 9x + 12 = 12x - 12 \] Now, we can solve for \( x \): \[ 12 + 12 = 12x - 9x \] \[ 24 = 3x \] \[ x = 8 \] Now substituting \( x = 8 \) back into the expression for \( m \angle Y X Z \): \[ m \angle Y X Z = 6(8) - 6 = 48 - 6 = 42 \] Thus, since \( m \angle W X Z = m \angle Y X Z \): \[ m \angle W X Z = 42 \] Therefore, the measure of angle \( W X Z \) is: \[ \boxed{42} \]