Answer:
Consider the triangle formed by points O, A, and P. Since OA and OP are radii of the circle centered at O, they are congruent. This makes triangle OAP isosceles, and since PA is tangent to the circle at A, angle OAP = 90 degrees.
Using the angle sum property in triangle OAP, we get:
∠OAP + ∠OAB + ∠PAB = 180°
Substituting 90° for ∠OAP and noting that angles at the points of tangency (like ∠PAB) are congruent, we get:
90° + ∠OAB + ∠PAB = 180°
Since we are given that ∠APB = 30°, we can replace ∠PAB with 30° in the equation.
90° + ∠OAB + 30° = 180°
Combining like terms, we get:
∠OAB + 120° = 180°
Subtracting 120° from both sides isolates ∠OAB:
∠OAB = 180° - 120°
Therefore, ∠OAB = 60°.