1. Let's first clarify the problem: we want to find the angle $\angle ABO$ in a given geometric figure where points A, B, and O are defined. 2. Typically, to find an angle like $\angle ABO$, we need the coordinates of points A, B, and O or the lengths of the sides of the triangle formed by these points. 3. If coordinates are given, we can use the vector approach: find vectors $\overrightarrow{BA}$ and $\overrightarrow{BO}$. 4. The angle $\angle ABO$ is the angle between vectors $\overrightarrow{BA}$ and $\overrightarrow{BO}$. 5. The formula for the angle $\theta$ between two vectors $\mathbf{u}$ and $\mathbf{v}$ is: $$\theta = \cos^{-1}\left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}\right)$$ 6. Calculate the dot product $\mathbf{u} \cdot \mathbf{v}$ and the magnitudes $\|\mathbf{u}\|$ and $\|\mathbf{v}\|$. 7. Substitute these values into the formula and compute $\theta$. 8. If side lengths are given instead, use the Law of Cosines: $$\cos(\angle ABO) = \frac{AB^2 + BO^2 - AO^2}{2 \cdot AB \cdot BO}$$ 9. Calculate the right side and then find $\angle ABO = \cos^{-1}(\text{value})$. 10. This method gives the measure of angle $\angle ABO$ in degrees or radians depending on your calculator settings.