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.