Since lines AC and BD intersect at point O, the angles ∠AOD and ∠BOC are vertical angles. Vertical angles are equal, so we can set the expressions for the angles equal to each other:
\[ m∠AOD = m∠BOC \]
Given that:
\[ m∠AOD = (5x - 4)° \] \[ m∠BOC = (4x + 5)° \]
We can set up the equation:
\[ 5x - 4 = 4x + 5 \]
Now, let's solve for \(x\):
- Subtract \(4x\) from both sides:
\[ 5x - 4x - 4 = 5 \]
This simplifies to:
\[ x - 4 = 5 \]
- Add 4 to both sides:
\[ x = 9 \]
Now that we have the value of \(x\), we can substitute it back to find \(m∠BOC\):
\[ m∠BOC = 4x + 5 \]
Substituting \(x = 9\):
\[ m∠BOC = 4(9) + 5 = 36 + 5 = 41° \]
Thus, the measure of angle \(BOC\) is:
\[ \boxed{41°} \]
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