https://file.aiquickdraw.com/mathbot/file/17264863746086bfq8zu9.jpg

In the image, we are given a circle with center \( O \), and a tangent line \( PQ \) at point \( A \). Several angles are marked within the circle:

- Angle \( \angle ACB = 15^\circ \)
- Angle \( \angle ABC = 30^\circ \)

We are asked to find the values of the angles \( x \), \( y \), and \( z \).

Here are some geometric principles that will help solve this:

1. **Angle between tangent and radius**: The angle between the tangent and the radius at the point of tangency is always \( 90^\circ \). So, \( \angle PAO = 90^\circ \).

2. **Angle subtended by the same arc (angle at the center and on the circumference)**: The angle subtended by an arc at the center is twice the angle subtended by the same arc at the circumference. This is useful for angles involving \( \angle ACB \) and \( \angle AOB \).

Let's proceed step by step to find \( x \), \( y \), and \( z \):

### Step 1: Finding \( x \)
Since \( \angle ABC = 30^\circ \), and this is an angle at the circumference, the angle subtended by the same arc at the center \( O \) will be twice that, so:
\[
x = 2 \times 30^\circ = 60^\circ
\]

### Step 2: Finding \( y \)
Similarly, \( \angle ACB = 15^\circ \), which is also at the circumference, so the angle subtended by the same arc at the center will be:
\[
y = 2 \times 15^\circ = 30^\circ
\]

### Step 3: Finding \( z \)
Since \( PQ \) is a tangent to the circle at point \( A \), \( \angle PAO = 90^\circ \). Angle \( z \) is the angle between the tangent line \( PQ \) and the chord \( AB \). From the exterior angle theorem in the cyclic quadrilateral, \( z \) is equal to the sum of the opposite interior angles in the triangle, i.e., \( \angle ABC \) and \( \angle ACB \).

Thus:
\[
z = 30^\circ + 15^\circ = 45^\circ
\]

### Final Answers:
- \( x = 60^\circ \)
- \( y = 30^\circ \)
- \( z = 45^\circ \)

Would you like a detailed explanation on any of the steps?

Here are some related questions to deepen your understanding:
1. How do we know the angle between a tangent and radius is \(90^\circ\)?
2. What is the cyclic quadrilateral theorem and how does it apply here?
3. Why is the angle subtended by an arc at the center twice that on the circumference?
4. What would happen if the triangle was not inscribed in the circle?
5. How would the result change if \( \angle ABC \) or \( \angle ACB \) were different?

**Tip**: Always remember that the sum of angles in a triangle adds up to \(180^\circ\), which helps in solving many geometry problems involving circles.

In the given figure, PQ is a tangent to the circle at A. Find the values of x, y, and z.

Solve the problem involving a tangent PQ to a circle at A, where angles at the circumference and center are provided. Use theorems such as the tangent-radius theorem and exterior angle theorem to find the values of x, y, and z. Suitable for Grades 9-10.

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