ABC is a triangle, BFC is a straight line and $\angle DEF$ is a right angle.

1. Identify the angle between the vertical segment EF and the horizontal base BC.

Since base $BC$ is horizontal and $EF$ is a vertical segment, $EF$ is perpendicular to $BC$.
Therefore, $\angle EFB = 90^\circ$.

2. Calculate $\angle BEF$ using the angle sum of triangle EFB.

In $\triangle EFB$:
$\angle BEF = 180^\circ - \angle EFB - \angle EBF$
$\angle BEF = 180^\circ - 90^\circ - 58^\circ = 32^\circ$