We are given a triangle inscribed in a circle, with $AC = BC$ (i.e., triangle $ABC$ is isosceles), and $\angle ABC = 72^\circ$. We are tasked with finding $x$, which represents $\angle ACB$.

Steps to Solve:

1. Since $AC = BC$, triangle $ABC$ is isosceles, so $\angle BAC = \angle ACB = x$.

2. The sum of the angles in any triangle must equal $180^\circ$. Therefore, in triangle $ABC$:

$\angle ABC + \angle BAC + \angle ACB = 180^\circ$

Substitute the given angle and the unknowns:

$72^\circ + x + x = 180^\circ$

3. Simplifying the equation:

$72^\circ + 2x = 180^\circ$

$2x = 180^\circ - 72^\circ = 108^\circ$

$x = \frac{108^\circ}{2} = 54^\circ$

Thus, $x = 54^\circ$.

Conclusion:

The measure of $\angle ACB$ is $54^\circ$.

Do you have any questions or would you like a further explanation?

Here are five related questions:

1. How do we calculate the angles in any triangle?
2. What are the properties of an isosceles triangle?
3. How do we use the sum of angles in a triangle to solve for unknowns?
4. What are the rules for angles in a cyclic quadrilateral?
5. How do we use geometric theorems to solve problems involving circles?

Tip: When solving problems involving isosceles triangles, remember that two angles are equal, which simplifies finding unknown angles.