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To solve this problem, we need to focus on the geometric configuration and the properties of tangents and triangles. Let's analyze the diagram and the information provided:

- \( A \), \( B \), \( C \), and \( D \) are points of tangency, meaning the lines connecting these points are tangent to the circles.
- The angle at \( A \) and \( D \) is given as \( 100^\circ \).
- We are asked to find the value of the angle \( x \), which is the angle at the top of the triangle, formed by tangents and the line segment \( AD \).

### Step-by-step approach:

1. **Angle of a tangent and radius**: When a line is tangent to a circle, the angle between the radius and the tangent line at the point of tangency is \( 90^\circ \). This helps us understand that some of the smaller triangles in the figure have right angles.

2. **Recognizing cyclic quadrilateral properties**: The figure forms a quadrilateral \( ABCD \), which is a cyclic quadrilateral (since the points of tangency lie on a common circle). A key property of cyclic quadrilaterals is that opposite angles sum to \( 180^\circ \). Specifically, \( \angle A + \angle D = 180^\circ \).

3. **Calculation**:
   - Given that \( \angle A = 100^\circ \), we can determine that \( \angle D = 80^\circ \) (since \( 180^\circ - 100^\circ = 80^\circ \)).
   - The angles at \( B \) and \( C \) are both right angles because the tangents to a circle meet the radius at \( 90^\circ \).
   
4. **Summing angles of the triangle**:
   - In \( \triangle ABC \), the sum of the interior angles must equal \( 180^\circ \). We already know two angles: \( \angle ABC = 90^\circ \) and \( \angle ACB = 90^\circ \).
   - Therefore, to find \( x \), the angle at the top of the triangle, we subtract from \( 180^\circ \):
   \[
   x = 180^\circ - 90^\circ - 40^\circ = 40^\circ.
   \]

Thus, the value of \( x \) is \( \boxed{40^\circ} \).

Let me know if you would like more details or have any questions.

Here are 5 related questions you can try:
1. What is the sum of the interior angles of any triangle?
2. How do tangents to a circle behave with respect to the radius at the point of tangency?
3. What is a cyclic quadrilateral, and what are its properties?
4. How does the tangent-secant angle theorem apply in geometric figures involving circles?
5. What is the relationship between opposite angles in cyclic quadrilaterals?

**Tip**: For problems involving circles, always look for properties related to tangents, radii, and cyclic quadrilaterals to simplify the solution.

De la figura, calcular el valor de x, si A, B, C y D son puntos de tangencia.

Learn how to calculate the angle x in a cyclic quadrilateral where points A, B, C, and D are points of tangency. This geometry problem covers key concepts such as tangents, cyclic quadrilaterals, and the sum of interior angles of triangles. Suitable for students in Grades 9-11.

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