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The problem provided involves a geometric figure where \( PA \) and \( QA \) are tangents to a circle at points \( X \) and \( Z \), respectively. The key given information includes:

- \( \angle XYZ = 30^\circ \), where \( Y \) is a point on the circle.
- We are tasked with finding \( \angle XAZ \) and \( \angle PAQ \).

### Step-by-step solution:

1. **Understanding Tangents**:
   - \( PA \) and \( QA \) are tangents to the circle. Therefore, \( \angle PAX = \angle QAZ = 90^\circ \) since a tangent is perpendicular to the radius at the point of tangency.

2. **Angle in a Circle Theorem (Inscribed Angle Theorem)**:
   - \( \angle XYZ \) is an inscribed angle, and it subtends arc \( XZ \). According to the inscribed angle theorem, the angle subtended by the same arc at the center of the circle (which would be \( \angle XAZ \)) is twice the inscribed angle:
     \[
     \angle XAZ = 2 \times \angle XYZ = 2 \times 30^\circ = 60^\circ
     \]

3. **Calculating \( \angle PAQ \)**:
   - The angle \( \angle PAQ \) can be computed as follows. Since \( PA \) and \( QA \) are tangents to the circle, the angle \( \angle PAQ \) is formed by the tangents. There's a known relation:
     \[
     \angle PAQ = 180^\circ - \angle XAZ
     \]
     Substituting the value of \( \angle XAZ \):
     \[
     \angle PAQ = 180^\circ - 60^\circ = 120^\circ
     \]

### Final Answers:
- \( \angle XAZ = 60^\circ \)
- \( \angle PAQ = 120^\circ \)

Let me know if you'd like further explanation or details!

### Relative Questions:
1. How do tangent lines relate to the radius of a circle at the point of tangency?
2. What is the inscribed angle theorem, and how does it apply to this problem?
3. Can you derive a general formula for the angle formed by two tangents to a circle?
4. How does the concept of supplementary angles help in finding \( \angle PAQ \)?
5. What other theorems are useful when dealing with tangents and circles?

### Tip:
Remember that angles subtended by the same arc on the circumference of a circle are equal. This concept is crucial in solving problems involving inscribed angles.

In the figure, PA and QA touch the circle at X and Z, respectively. Y is a point on the circle such that ∠XYZ = 30°. Find ∠XAZ and ∠PAQ.

Learn how to find ∠XAZ and ∠PAQ when tangents PA and QA touch a circle at points X and Z, with ∠XYZ = 30°. This problem involves key geometric theorems like the Inscribed Angle Theorem and the Tangent-Radius Perpendicularity. Suitable for students in Grades 10-12.

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