Answer
$m \ \widehat{BC}=142^{\circ}$
Work Step by Step
We must know that the measure of a minor arc is equal to the measure of its corresponding central angle.
In the given question, we can see that the corresponding angle of minor 6.0 pt$\widehat{BC}$ is $\angle BPC$ .
We have: $m \ \widehat{BC}=m \angle BPC$
Therefore, we can write
$m\angle BPC=360^{\circ}-90^{\circ}-128^{\circ}=142^{\circ}$
Thus, $m \ \widehat{BC}=142^{\circ}$
