To find the angle \( \angle OAB \) in the given diagram, let's use the information we've already determined.

### Key Information:
- \( \angle OBC = 90^\circ \) because \( OB \) is the radius and \( AB \) is the tangent.
- \( \angle OCB = 64^\circ \) (as given in the problem).
- \( \angle ABC = 154^\circ \) (calculated previously).

### To find \( \angle OAB \):
\( \angle OAB \) is an external angle to triangle \( OAB \), and it equals the difference between \( \angle ABC \) and \( \angle OBC \). The reasoning is:

- \( \angle OAB = \angle ABC - \angle OBC \)
- \( \angle OAB = 154^\circ - 90^\circ = 64^\circ \)

So, \( \angle OAB = 64^\circ \).

This result is consistent with the fact that \( \angle OCB \) and \( \angle OAB \) are equal because they subtend the same arc \( BC \) of the circle (this follows from the Inscribed Angle Theorem and its corollary related to angles in the same segment).

Would you like more details on this result, or have any other questions? Here are some related questions:

1. What is the Inscribed Angle Theorem and how is it used here?
2. How do we know that \( \angle OAB \) and \( \angle OCB \) are equal?
3. Can we determine the length of side \( AB \) using trigonometry?
4. What if the circle had a different radius—how would the angles change?
5. What properties of tangents are applied in this problem?

**Tip**: In circle geometry, external angles formed by tangents often give key insights into solving problems, especially when using properties like the Inscribed Angle Theorem.

Learn how to calculate angle OAB in circle geometry with a detailed step-by-step solution. Understand the Inscribed Angle Theorem and its application in determining angles formed by tangents and chords. Suitable for high school students studying geometry.

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