To solve the given problem, we will analyze the geometry of the circle and the relationships between angles.

Problem Analysis:

• $\angle BOA = 140^\circ$
• $\angle OAC = 50^\circ$
• We are tasked with finding $\angle OBC$.

Step 1: Recognize Relationships in the Circle

1. $O$ is the center of the circle, so $OA$, $OB$, and $OC$ are radii. This makes triangles $\triangle OAB$ and $\triangle OAC$ isosceles.
2. The angle subtended by an arc at the center is twice the angle subtended at the circumference by the same arc.

Step 2: Find $\angle ABC$

1. The angle subtended at $A$ and $C$ (by arc $BC$) can be related as follows: \angle BOA = 140^\circ \implies \text{Arc BC} \text{ subtends an angle of } \frac{140^\circ}{2} = 70^\circ \text{ at the circumference}. Thus, $\angle ABC = 70^\circ$.

Step 3: Calculate $\angle OBC$

1. Since $\triangle OAC$ is isosceles ($OA = OC$), $\angle OAC = 50^\circ$, and $\angle OCA = 50^\circ$ as well.
2. Use the fact that $\triangle ABC$ is a triangle: [ \angle OBC = \angle ABC - \