1. **Problem statement:** Given a circle with a triangle inscribed such that one angle is $46.8^\circ$ and the angle $x$ is on the opposite side of the diameter line, find the value of $x$. 2. **Key concept:** In a circle, angles subtended by the same chord and on the same segment are equal. Also, angles on opposite sides of a diameter are supplementary (sum to $180^\circ$) if they form a straight line. 3. Since the triangle is inscribed in the circle and $x$ and $46.8^\circ$ are on opposite sides of the diameter, they are equal because they subtend the same arc. 4. Therefore, $x = 46.8^\circ$. 5. **Answer:** $\boxed{46.8^\circ}$