To solve this question, we need to calculate the angle $x$ of the kite inside the regular octagon. Here's a detailed step-by-step explanation:

1. Properties of a Regular Octagon:

   • A regular octagon has 8 equal sides and angles.
   • The sum of the interior angles of a polygon is given by: $\text{Sum of interior angles} = (n-2) \times 180^\circ$ where $n$ is the number of sides. For an octagon ($n = 8$): $\text{Sum of interior angles} = (8-2) \times 180^\circ = 1080^\circ$
   • Each interior angle of a regular octagon is: $\text{Interior angle} = \frac{1080^\circ}{8} = 135^\circ$

2. Analyzing the Kite Inside the Octagon:

   • The kite's apex angle ($x$) at the center of the octagon is formed by two adjacent sides of the octagon.
   • These adjacent sides make an angle equal to the interior angle of the octagon, which is $135^\circ$.

3. Conclusion:

   • Therefore, the value of $x$ is: $x = 135^\circ$

If you have any further questions or would like clarification, feel free to ask!

Related Questions:

1. What is the formula for the sum of interior angles of any polygon?
2. How would the calculation change if the polygon was a hexagon instead of an octagon?
3. What are the exterior angles of a regular octagon, and how do they relate to the interior angles?
4. How do you calculate the area of a regular octagon if the side length is given?
5. Can this kite’s properties be extended to other polygons? Why or why not?

Tip: Always start solving such problems by recalling the key properties of the polygon involved—this makes the calculations straightforward!