Question 1210494
The value of $x$, which is $\angle AGB$, is $\mathbf{120^\circ}$.

Here is the reasoning based on the properties of an equilateral triangle and its centroid.

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## Centroid and Medians in an Equilateral Triangle

1.  **Medians are also Angle Bisectors and Altitudes:** In an **equilateral triangle** ($\triangle ABC$), the medians (which intersect at the centroid $G$) are also the angle bisectors and the altitudes.
2.  **Symmetry:** Because $\triangle ABC$ is equilateral, it has three-fold rotational symmetry. This means that the area, angles, and distances formed by the centroid to the three vertices are identical.
3.  **Angles around the Centroid:** The centroid $G$ divides the triangle into three smaller triangles: $\triangle AGB$, $\triangle BGC$, and $\triangle CGA$. The angles formed at the centroid ($\angle AGB$, $\angle BGC$, and $\angle CGA$) must sum to $360^\circ$.
    $$\angle AGB + \angle BGC + \angle CGA = 360^\circ$$
4.  **Equal Angles:** Due to the symmetry of the equilateral triangle, these three central angles must be equal:
    $$\angle AGB = \angle BGC = \angle CGA$$

### Calculation

Let $x = \angle AGB$.
$$x + x + x = 360^\circ$$
$$3x = 360^\circ$$
$$x = \frac{360^\circ}{3} = \mathbf{120^\circ}$$

Therefore, $\angle AGB = 120^\circ$.