Thank you for the clarification. Let's revisit the problem.

Given:

1. $DE$ is a tangent to the circle at $B$.
2. $AB = AC$ (triangle $ABC$ is isosceles).
3. $\angle ABE = 48^\circ$ (angle between tangent $DE$ and chord $AB$).

Applying the Alternate Segment Theorem

The Alternate Segment Theorem states that the angle between the tangent and a chord through the point of contact is equal to the angle in the alternate segment. Therefore, $\angle ABE = \angle ACB$.

Given: $\angle ABE = 48^\circ$

So: $\angle ACB = 48^\circ$

Thus, the value of $x$ is $48^\circ$.

Would you like further details or explanations? Here are some questions related to the concept:

1. What is the Alternate Segment Theorem?
2. How do you apply the isosceles triangle property in circle geometry?
3. What are the key properties of a tangent to a circle?
4. How does the inscribed angle relate to the arc it subtends?
5. What are other applications of circle theorems in geometry?

Tip: When dealing with circle theorems, always check if the problem involves the Alternate Segment Theorem or the angle subtended by the same arc, as they are often key to finding unknown angles.