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To solve for the value of \( x \) in the given circle diagram, we can apply the property of angles in a cyclic quadrilateral and other geometric rules involving circles.

Here’s the breakdown:

1. The angle of 55° and 48° are inscribed angles.
2. The angle \( x \) is formed by the intersection of two chords, and it lies opposite the angle of 48° in the cyclic quadrilateral.

### Key Properties to Apply:
- In a cyclic quadrilateral (a four-sided figure where all vertices lie on the circumference of a circle), the opposite angles sum to 180°.

Let's calculate:

\[
55^\circ + x = 180^\circ \quad \text{(opposite angles in a cyclic quadrilateral)}
\]

Solving for \( x \):

\[
x = 180^\circ - 55^\circ
\]
\[
x = 125^\circ
\]

Thus, the value of \( x \) is **125°**.

Would you like further clarification on this? Here are five related questions you can explore:

1. How can the properties of a cyclic quadrilateral be derived from circle theorems?
2. What is the relationship between inscribed angles and central angles in a circle?
3. How does the sum of angles in any quadrilateral differ from that of a cyclic quadrilateral?
4. Can you solve for unknown angles in more complex cyclic quadrilaterals?
5. What other geometric properties can be derived from angles subtended by the same arc?

**Tip**: Remember that angles subtended by the same arc in a circle are equal!

What is the value of x in a circle given angles 55° and 48° as inscribed angles?

Solve for the unknown angle x in a cyclic quadrilateral using circle theorems and properties of angles. This problem involves key concepts in circle geometry, including the cyclic quadrilateral theorem. Suitable for students in Grades 8-10.

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