https://file.aiquickdraw.com/mathbot/file/172375618605333vt047g.jpg

To solve for angles \( x \) and \( y \) in the given diagram, we will use the properties of circles and angles subtended by arcs.

### Step 1: Understanding the diagram
- \( \angle 55^\circ \) is an interior angle within the circle.
- \( \angle 70^\circ \) is the exterior angle at point \( A \), formed between the tangent \( AT \) and the chord \( AB \).
- \( x \) and \( y \) are the angles that need to be determined.

### Step 2: Finding angle \( x \)
Angle \( x \) is an angle at the circumference subtended by the same arc as the angle \( 55^\circ \). According to the circle theorem, angles subtended by the same arc on the circumference are equal. 

Therefore, 
\[
x = 55^\circ
\]

### Step 3: Finding angle \( y \)
Angle \( y \) is formed by the tangent \( AT \) and the chord \( AB \). According to the alternate segment theorem, the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Therefore, angle \( y \) is equal to angle \( x \):

\[
y = 55^\circ
\]

However, considering that \( \angle 70^\circ \) is an exterior angle formed by the tangent and the chord \( AB \), \( y \) can also be found as:

\[
y = 180^\circ - 70^\circ = 110^\circ
\]

Thus, angle \( y = 110^\circ \).

### Final Answers:
- \( x = 55^\circ \)
- \( y = 110^\circ \)

Would you like further details or have any other questions?

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Here are 8 follow-up questions related to this problem:
1. What is the circle theorem that explains why \( x = 55^\circ \)?
2. How do we apply the alternate segment theorem in this scenario?
3. What is the sum of angles in any triangle, and how does it relate to this problem?
4. Can you explain why the exterior angle \( y \) is calculated as \( 180^\circ - 70^\circ \)?
5. How would the solution change if the angle at \( A \) was different?
6. What are other examples where the alternate segment theorem is applicable?
7. Can you determine the angle subtended at the center by the same chord?
8. How would the problem change if the tangent was on the opposite side of \( A \)?

### Tip:
In geometry, always consider all the theorems that might apply to a situation. Sometimes, different theorems can give the same answer, reinforcing your solution.

Learn how to solve for angles x and y in a circle using circle theorems and the alternate segment theorem. This geometry problem involves calculating angles subtended by arcs and formed by tangents and chords.

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