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.