The image shows a circle with a triangle inscribed within it. Let’s walk through the solution to find the value of $x^\circ$.

Observations:

1. There is a point $O$, likely the center of the circle.
2. One vertex of the triangle is at the center $O$.
3. The two other vertices lie on the circumference, forming an isosceles triangle with two radii of the circle.
4. The angle marked $x^\circ$ is at the circumference.

Since the angle is formed on the circle (subtended at the circumference) and the larger central angle subtends the same arc, we can apply the angle at the center theorem.

Key Theorem:

• Angle at the center is twice the angle at the circumference when subtending the same arc.

Applying the Theorem:

If the central angle (the angle at $O$) subtends the same arc, it is twice the value of $x^\circ$.

$2x = 180^\circ \quad (\text{as the angle at } O \text{ is } 180^\circ)$

$x = \frac{180^\circ}{2} = 90^\circ$

Thus, the value of $x^\circ$ is:

$\boxed{90^\circ}$

Would you like further details or have any specific questions?

Here are 5 related questions to deepen your understanding:

1. What is the angle subtended by a diameter at the circumference of a circle?
2. How does the angle at the center theorem apply in quadrants of a circle?
3. Can the angle at the center theorem be generalized to non-isosceles triangles?
4. How does this problem change if the central angle is not 180 degrees?
5. What other properties of circles involve angles on the circumference?

Tip: Always look for isosceles triangles when you see radii or diameters involved in a circle problem.