Given Information:
1. $O$ is the center of the circle.
2. $\(\angle A C O=40^{\circ}\)$.

Goal: Find the degree measure of $\angle A B C$.

Step-by-Step Solution:

1. Identify Radii and Isosceles Triangle:

- Since $O$ is the center of the circle, segments $O A, O B$, and $O C$ are all radii.
- Therefore, $O A=O C$.
- This means that $\triangle A O C$ is an isosceles triangle.

2. Find Base Angle of $\(\triangle A O C\)$ :

- In an isosceles triangle, the angles opposite the equal sides are equal.
- We are given $\(\angle A C O=40^{\circ}\)$.
- So, $\(\angle C A O=\angle A C O=40^{\circ}\)$.

3. Calculate the Minor Central Angle $\(\angle A O C\)$ :

- The sum of angles in a triangle is $180^{\circ}$.
- In $\triangle A O C$, the angle at the center is $\(\angle A O C=180^{\circ}-(\angle C A O+\angle A C O)\)$
- $\(\angle A O C=180^{\circ}-\left(40^{\circ}+40^{\circ}\right)\)$
- $\(\angle A O C=180^{\circ}-80^{\circ}\)$
- $\\(angle A O C=100^{\circ}\)$.
- This $\(100^{\circ}\)$ is the measure of the minor central angle subtending the minor arc AC .

4. Calculate the Reflex Central Angle $\(\angle A O C\)$ :

- A full circle measures $\(360^{\circ}\)$.
- The reflex central angle $\angle A O C$ (the larger angle around O that subtends the major arc AC ) is $\(360^{\circ}-\)$ minor $\(\angle A O C\)$.
- Reflex $\(\angle A O C=360^{\circ}-100^{\circ}=260^{\circ}\)$.

5. Relate Inscribed Angle $\(\angle A B C\)$ to the Reflex Central Angle:

- An inscribed angle is half the measure of the central angle that subtends the same arc.
- In the figure, $\(\angle A B C\)$ is an inscribed angle. If it is interpreted as subtending the major arc AC (the arc that passes through B ), then its measure is half of the reflex central angle $\(\angle A O C\)$.
- $\(\angle A B C=\frac{1}{2} \times\)$ Reflex $\angle A O C$.

6. Calculate $\angle A B C$ :
- $\(\angle A B C=\frac{1}{2} \times 260^{\circ}\)$
- $\(\angle A B C=130^{\circ}\)$.