To find the value of $x$ in the given circle diagram, we can use the property of intersecting chords within a circle.

1. Theorem Used: The measure of an angle formed by two intersecting chords (like $\angle B$) is equal to half the sum of the measures of the arcs intercepted by the angle and its opposite angle.

2. Given Information:

   • $\angle ABE = 77^\circ$
   • $\angle CBD = 55^\circ$

3. Solution: Since $\angle ABE$ and $\angle CBD$ are opposite angles formed by intersecting chords, the angle opposite $\angle ABE$ (the arc $x$ in this case) will have a measure based on the sum of $\angle ABE$ and $\angle CBD$.

   Using the formula: $x = \angle ABE + \angle CBD$ Substituting in the values: $x = 77^\circ + 55^\circ = 132^\circ$

So, the value of $x$ is $132^\circ$.

Would you like a more detailed explanation, or do you have any questions?

Here are five related questions to deepen understanding:

1. How do we calculate the angle at a point where two chords intersect within a circle?
2. What other theorems apply to angles and arcs within a circle?
3. How would the solution change if $\angle ABE$ and $\angle CBD$ were different values?
4. Can this theorem be applied to find angles outside the circle?
5. How do we find the measure of an angle if only one intercepted arc is known?

Tip: Remember, for any two intersecting chords, the angle between them is always half the sum of the measures of the intercepted arcs.