To find the measure of arc $DC$, let's analyze the given information:

1. In the diagram, $\angle DAC = 58^\circ$ and $\angle DCA = 106^\circ$.
2. We observe that $D$, $A$, $C$, and $O$ form a cyclic quadrilateral within a circle.

For cyclic quadrilaterals, opposite angles add up to $180^\circ$. Therefore:

$\angle DAC + \angle DCA = 180^\circ$

Since $\angle DAC = 58^\circ$, it implies:

$\angle ADC = 180^\circ - 106^\circ = 74^\circ$

Since