To solve the given problem, let's carefully apply the circle theorems and properties.

Key Information:

1. Circle Properties:

   • The angle between the tangent and a chord at the point of contact is equal to the angle in the alternate segment (alternate segment theorem).
   • The sum of the angles in a triangle is $180^\circ$.

2. Given Values:

   • $\angle BAC = 23^\circ$,
   • $\angle ACB = 71^\circ$.

We are tasked to find the value of $x^\circ$.

Steps to Solve:

1. Use the Alternate Segment Theorem:

   • The angle between the tangent and the chord ($\angle x$) is equal to the angle in the alternate segment, which is $\angle ACB = 71^\circ$.

   Therefore: $x = 71^\circ$

Conclusion:

The value of $x^\circ$ is: $\boxed{71^\circ}$

Let me know if you'd like further clarifications or detailed explanations about the theorems used!

Related Questions:

1. How does the alternate segment theorem work in circles?
2. Can the value of $x$ change if the positions of $A$, $B$, or $C$ were altered?
3. What other angle properties of circles could be applied in similar problems?
4. How can tangents to a circle help solve geometric problems?
5. Could this problem be extended with the radius of the circle?

Tip: Always identify known relationships, such as the tangent and alternate segment theorem, when solving circle geometry problems.