The problem asks to find the value of $x$ in the given circle with center $O$. We are given that the angle at the center subtended by one arc is $80^{\circ}$, and the angle at a point on the circumference subtended by another arc is $60^{\circ}$. The angle $x$ is the angle subtended at the circumference by the same arc as the $80^{\circ}$ angle at the center.

1. Problem Description

The problem asks to find the value of $x$ in the given circle with center $O$. We are given that the angle at the center subtended by one arc is $80^{\circ}$, and the angle at a point on the circumference subtended by another arc is $60^{\circ}$. The angle $x$ is the angle subtended at the circumference by the same arc as the $80^{\circ}$ angle at the center.

2. Solution Steps

The angle at the center is twice the angle at the circumference subtended by the same arc.

Angle at the center = $2 \times$ Angle at the circumference

We are given that the angle at the center is $80^{\circ}$.

We have to find the angle $x$ at the circumference subtended by the same arc.

$80^{\circ} = 2 \times x$

Divide both sides by 2:

$x = \frac{80^{\circ}}{2}$

$x = 40^{\circ}$

3. Final Answer

$x = 40^{\circ}$