This is a geometry problem involving a cyclic triangle inscribed in a circle. The given angle of $51^\circ$ and the angle $x$ are both inscribed angles subtended by the same arc of the circle. According to the properties of a cyclic quadrilateral, or more specifically the "Angles Subtended by the Same Arc" theorem, angles subtended by the same arc are equal.

Since both the angle $51^\circ$ and $x$ subtend the same arc, we can conclude that:

$x = 51^\circ$

Therefore, the value of $x$ is:

$x = 51^\circ$

Would you like more details on this solution, or do you have further questions?

Here are 5 related questions to explore:

1. What are the properties of cyclic quadrilaterals?
2. How do inscribed angles relate to the arcs they subtend?
3. What is the difference between central and inscribed angles?
4. Can this property be applied to polygons with more than three sides?
5. How do we calculate angles in circles when given multiple intersecting chords?

Tip: Always remember that inscribed angles subtended by the same arc are equal! This is a key property in solving many circle-related geometry problems.