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.