In this problem, we are given:

• $\angle CAP = 34^\circ$
• $S$ is the incenter of $\triangle ABC$

Since $S$ is the incenter, it is the point where the angle bisectors of $\triangle ABC$ meet. Therefore:

1. $\angle CAP$ is half of $\angle CAB$ because $AP$ is an angle bisector of $\angle CAB$.

To find $\angle CAB$: $\angle CAB = 2 \times \angle CAP = 2 \times 34^\circ = 68^\circ$

So, the correct answer is: $\angle CAB = 68^\circ$

Would you like further details or have any questions?

Here are some additional related questions for further exploration:

1. How can the incenter property be used to find other angle measures in a triangle?
2. If the incenter divides another angle in the triangle, what relationships exist between the divided angles?
3. How does knowing one angle bisector affect understanding of a triangle’s properties?
4. What properties define an incenter and differentiate it from other triangle centers, like the centroid?
5. How would the solution differ if $S$ were the centroid instead of the incenter?

Tip: Remember, the incenter is equidistant from each side of the triangle and is the point of intersection for all angle bisectors.