Since ray BE bisects angle ABC, we have:
\[ m\angle ABE = m\angle CBE \]
According to the problem, we have:
\[ m\angle ABE = (2x + 7)^\circ \] \[ m\angle CBE = (5x - 8)^\circ \]
Setting the two expressions equal to each other gives us:
\[ 2x + 7 = 5x - 8 \]
Now, we solve for \(x\):
- Subtract \(2x\) from both sides:
\[ 7 = 3x - 8 \]
- Add \(8\) to both sides:
\[ 15 = 3x \]
- Divide by \(3\):
\[ x = 5 \]
Now we substitute \(x\) back into the expression for \(m\angle ABE\):
\[ m\angle ABE = 2(5) + 7 = 10 + 7 = 17^\circ \]
Thus, the measure of \(m\angle ABE\) is:
\[ \boxed{17} \]