We are given a circle with an inscribed angle. One intercepted arc has a measure of $120^\circ$. We need to find the measure of angle $x$, where $x$ is formed by the intersection of a chord and a diameter.

1. Problem Description

We are given a circle with an inscribed angle. One intercepted arc has a measure of $120^\circ$. We need to find the measure of angle $x$, where $x$ is formed by the intersection of a chord and a diameter.

2. Solution Steps

The inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.

So, if the intercepted arc is $120^\circ$, the inscribed angle is $\frac{120^\circ}{2} = 60^\circ$.

The angle $x$ is supplementary to the inscribed angle, meaning that their sum is $180^\circ$.

Let the inscribed angle be $A$. We found that $A = 60^\circ$.

Then $x + A = 180^\circ$.

$x + 60^\circ = 180^\circ$.

Subtracting $60^\circ$ from both sides gives:

$x = 180^\circ - 60^\circ$.

$x = 120^\circ$.

Since the angle $x$ seems to be on the major arc of the inscribed angle, we can calculate angle $x$ as follows:

The measure of the inscribed angle is half of the intercepted arc, so the measure of the inscribed angle is $\frac{120}{2} = 60$ degrees.

The straight line segment passing through the center of the circle creates a diameter. The measure of the angle formed by the diameter (a straight line) is 180 degrees.

Angle $x$ is supplementary to the inscribed angle. Therefore, angle $x$ is $180 - 60 = 120$ degrees.

3. Final Answer

120