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We are given a circle with center \( O \), and points \( A \), \( B \), and \( C \) lying on the circumference. The chord \( BA \) is extended to point \( T \), making \( CT \) tangent to the circle at \( C \). We are also given that:

- \( \angle ATC = 30^\circ \)
- \( \angle ACT = 50^\circ \)

We need to find the angle \( \angle BOA \).

### Steps to Solve:

1. **Using Tangent-Secant Theorem**:  
   Since \( CT \) is tangent to the circle at \( C \), and \( ATC \) forms a triangle, we can use the fact that the angle subtended by a tangent and a chord is equal to the angle subtended by the same chord at the circumference of the circle.

   Thus:
   \[
   \angle ACT = \angle BAC = 50^\circ
   \]

2. **Angle in the Circle**:  
   The angle \( \angle BOA \) is the central angle subtended by arc \( BA \), and we know that the central angle is twice the angle subtended by the same arc at the circumference.

   So:
   \[
   \angle BOA = 2 \times \angle BAC = 2 \times 50^\circ = 100^\circ
   \]

Thus, the angle \( \angle BOA \) is \( 100^\circ \).

### Answer:
The correct option is:
\[
\boxed{100^\circ}
\]

Would you like further details or have any questions?

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Here are 5 related questions to expand on this concept:

1. How do you determine angles subtended by the same arc in a circle?
2. What is the tangent-secant angle theorem, and how is it applied?
3. How can we determine the angle between a chord and a tangent in a circle?
4. What are the properties of central and inscribed angles in a circle?
5. How does the concept of cyclic quadrilaterals relate to circle theorems?

**Tip:** Remember that the central angle subtended by an arc is always twice the angle subtended by the same arc on the circumference.

In the figure, A, B, and C are three points on a circle with center O. The chord BA is extended to a point T such that CT becomes a tangent to the circle at point C. If ∠ATC = 30° and ∠ACT = 50°, then the angle ∠BOA is?

This problem involves circle geometry, specifically finding the central angle ∠BOA using the tangent-secant theorem and the central angle theorem. It uses key formulas and theorems, such as the angle subtended by a chord and the central angle being twice the angle at the circumference. Ideal for students in Grades 9-11.

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