1. The problem asks to find the value of $x$ in the circle with center $O$ and points $A, B, C, D$ on the circumference. 2. We know that the angle at the center $O$ subtended by chord $AB$ is twice the angle at the circumference subtended by the same chord. This is a key property of circles: the central angle is twice the inscribed angle on the same arc. 3. Given the angle at point $B$ inside the circle is $82^\circ$, this is the inscribed angle subtended by chord $AB$. 4. Therefore, the central angle $x$ at $O$ subtended by chord $AB$ is: $$x = 2 \times 82^\circ = 164^\circ$$ 5. Hence, the value of $x$ is $164^\circ$ because the central angle is twice the inscribed angle subtending the same chord.