To solve the problem, we need to analyze the given diagram based on the provided information about the angles and the properties of the triangle.

### Given:
- \( AO \) is the angle bisector of angle \( BAD \).
- \( \angle OAB = \angle AOD \).

### To Prove:
1. \( \angle OAB = \angle AOD \): Since \( AO \) is the angle bisector, this condition is already satisfied.

2. To show that \( \frac{AB}{AD} = \frac{OB}{OD} \): By the Angle Bisector Theorem, we can state that the ratio of the lengths of the two segments created by the angle bisector is equal to the ratio of the lengths of the two other sides of the triangle.

### Steps:
1. **Identify Angles:**
\[
\angle OAB = \angle AOD
\]
This implies that the two angles are equal, confirming that \( AO \) bisects \( \angle BAD \).

2. **Use the Angle Bisector Theorem**:
Apply the Angle Bisector Theorem:
\[
\frac{AB}{AD} = \frac{OB}{OD}
\]
From the initial conditions, we can confirm that as \( AO \) is the bisector, the relationship holds.

3. By understanding that angles and sides behave proportionally in triangles, we conclude that the statement is valid.

So, the relationship between the segments created by the angle bisector is established through the properties of angles in triangles.