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To find the angle \( \theta \) in this configuration, we can observe that it involves two tangent circles within an isosceles triangle, with radii given as 4 and 20.

Here's the approach to solve this problem:

1. **Analyze the Geometry:**
   - The two circles are tangent to each other and to the sides of the triangle, implying that the centers of these circles lie along a straight line with the vertex of the angle \( \theta \) at one end.
   - We can set up a right triangle using the radii and the distance between the centers of the two circles.

2. **Calculate Distance Between Centers:**
   - The centers of the two circles are separated by the sum of their radii, which is \( 4 + 20 = 24 \) units.

3. **Use the Tangent Ratio:**
   - The distance between the centers (24 units) forms the adjacent side to the angle \( \theta \), while the difference in the radii (20 - 4 = 16 units) forms the opposite side of the right triangle.

4. **Calculate \( \theta \) using Trigonometry:**
   - We can use the tangent function, where:
     \[
     \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{16}{24}
     \]
   - Simplifying, \( \tan(\theta) = \frac{2}{3} \).

5. **Find \( \theta \):**
   - Using the arctangent function, calculate \( \theta \):
     \[
     \theta = \tan^{-1} \left(\frac{2}{3}\right)
     \]

Calculating this gives:

\[
\theta \approx 33.7^\circ
\]

**Answer:** The angle \( \theta \) is approximately \( 33.7^\circ \), rounded to the nearest tenth of a degree.

Would you like more details or have any questions?

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Here are 5 additional questions to expand your understanding:

1. How would the angle \( \theta \) change if the radii of the circles were different?
2. What would happen to \( \theta \) if the circles were not tangent to the triangle's sides?
3. How can this method be applied if there are more circles arranged in a similar manner?
4. What trigonometric functions can be used if we had a different known angle or side length?
5. Can this method be used to find angles in other types of polygons?

**Tip:** Always double-check the radius and distances to ensure accurate calculations, especially when applying trigonometric functions.

Find the value of the angle θ in degrees rounded to the nearest tenth of a degree.

Learn how to find the angle θ in a geometric configuration with tangent circles in an isosceles triangle. This problem uses trigonometric ratios and tangent functions, suitable for high school students.

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