Answer
Both are correct.
Work Step by Step
The definition of an angle bisector tells us that:
$m\angle ABX=m\angle CBX$
The angle addition postulate tells us that:
$m\angle ABX+m\angle CBX=m\angle ABC$
Substitute:
$m\angle ABX+m\angle ABX=m\angle ABC$
Combine like terms:
$2m\angle ABX=m\angle ABC\longrightarrow$ one student is correct
Divide each side by 2:
$2m\angle ABX\div2=m\angle ABC\div2$
$m\angle ABX=\frac{1}{2}m\angle ABC\longrightarrow$ the other student is also correct
