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In the following diagram, calculate the lettered angles.
A = 
 
° B = 
 
° C = 
 
°
 
Identify the circle theory.

To solve for the angles \( A \), \( B \), and \( C \) in the given diagram, we'll apply the **Alternate Segment Theorem** in circle geometry. The Alternate Segment Theorem states that the angle between the tangent and a chord through the point of contact is equal to the angle in the alternate segment of the circle.

Given:
- \(\angle C = 50^\circ\)
- \(\angle B = 53^\circ\)

### Steps:

1. **Angle \( A \):**
   \[
   \angle A = \angle BCT = 53^\circ
   \]
   (By the Alternate Segment Theorem, \(\angle A\) is equal to the angle between the tangent and chord at \( B \).)

2. **Angle \( B \):**
   \[
   \angle B = \angle CAT = 50^\circ
   \]
   (By the same theorem, \(\angle B\) is equal to the angle between the tangent and chord at \( C \).)

3. **Angle \( C \):**
   \[
   \angle C = 180^\circ - (\angle A + \angle B) 
   \]
   Since the sum of the angles in a triangle is 180°:
   \[
   \angle C = 180^\circ - (53^\circ + 50^\circ) = 77^\circ
   \]

### Conclusion:

- \( A = 53^\circ \)
- \( B = 50^\circ \)
- \( C = 77^\circ \)

### Circle Theory Used:
- **Alternate Segment Theorem**: This theorem was used to determine the angles \( A \) and \( B \).

Would you like any more details or have any questions?

### Related Questions:
1. How would you calculate angle \( C \) if angle \( A \) was known to be \( 60^\circ \) instead of \( 53^\circ \)?
2. Can you explain why the sum of angles in any triangle is always \( 180^\circ \)?
3. How does the Alternate Segment Theorem apply when a triangle is inscribed in a circle?
4. What are other important theorems related to angles in a circle?
5. If the tangent at \( B \) is extended, what would be the angle between the extended tangent and the chord \( AC \)?

### Tip:
Remember that the Alternate Segment Theorem can be a powerful tool in solving problems involving tangents and chords in a circle.

In the following diagram, calculate the lettered angles. A = __ °, B = __ °, C = __ °. Identify the circle theory.

Learn how to calculate angles in a triangle inscribed in a circle with tangents. This problem utilizes the Alternate Segment Theorem and is suitable for students in Grades 8-10. Explore the relationships between angles in circles and tangents.

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