https://file.aiquickdraw.com/mathbot/file/1725436473354nt81lgtn.jpg

The problem is asking to calculate the angle \(\beta\) given the following information from the diagram:

- \(t\) is a tangent to the circle at point \(C\).
- \(\alpha = 27^\circ\) is the angle between the line \(g_1\) and line \(g\).
- You are to find the angle \(\beta\) between line \(g_2\) and line \(g\).

### Steps to solve:

1. **Understanding the Geometry**:
   - \(t\) is tangent to the circle at point \(C\).
   - \(g_1\) and \(g_2\) are lines intersecting the circle.
   - \(M\) is the center of the circle.
   - \(g\) is a line intersecting lines \(g_1\) and \(g_2\) at points \(A\) and \(B\) respectively.

2. **Identify the given angles**:
   - The angle \(\alpha\) is the angle formed by line \(g_1\) and line \(g\) at point \(A\).
   - \(\beta\) is the angle between line \(g_2\) and line \(g\) at point \(B\).

3. **Use of the Tangent-Secant Theorem**:
   - The angle \(\angle tCM\) is the angle between the tangent and the radius (which is a right angle, \(90^\circ\)).
   - The angle \(\alpha\) formed by the line \(g_1\) with the tangent can be related to the angle between the tangent and the radius using the properties of circles and tangents.
   - The angle \(\beta\) can then be derived by understanding the relationship between the angles \(\alpha\) and \(\beta\) and the geometry of the circle.

4. **Calculation**:
   - Since \(M\) is the center, \( \angle AMC = \alpha \).
   - Using the alternate segment theorem, \( \angle MCA = \angle \beta \).
   - Given that \(\alpha = 27^\circ\), the corresponding \(\beta\) would also be \(27^\circ\), as the two angles in question form corresponding angles with respect to the tangent at \(C\).

Thus, \(\beta = 27^\circ\).

### Conclusion
The value of \(\beta\) is \(27^\circ\).

Would you like further details on any part of this solution, or have any other questions?

Here are some related questions you might find helpful:
1. How does the alternate segment theorem apply in this scenario?
2. What is the tangent-secant theorem and how is it used in circle geometry?
3. How are angles subtended by the same chord related in a circle?
4. Can you provide a proof of the alternate segment theorem?
5. What are the properties of angles formed by tangents and secants in a circle?

**Tip**: In circle geometry, remember that angles subtended by the same arc at the circumference are equal. This can help in solving various problems involving circles.

AM = Radius des Kreises g1 // g2, t ist Tangente an den Kreis im Punkt C, α = 27° Berechnen Sie den Winkel β

This circle geometry problem involves calculating the angle β given the tangent at point C and angle α = 27°. The solution uses the Tangent-Secant Theorem and Alternate Segment Theorem, making it suitable for students in Grades 9-12.

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