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The diagram you provided shows a triangle with labeled sides and angles. Let me describe the setup before solving for angle \( n \).

- You have a large triangle with a smaller triangle nested inside it.
- The larger triangle has two equal sides of length \( 2a \) and a base \( a + n \).
- The smaller triangle inside has sides of length \( a \) and an angle of \( \alpha \).
- Both triangles appear to be isosceles, with the two marked sides (on both the large and small triangles) being equal.

### Steps to find \( \angle n \):

1. **Isosceles Triangle Property:**
   The base angles of an isosceles triangle are equal. Therefore, in both the smaller triangle and the larger triangle, the two angles opposite the equal sides are equal.

2. **Angle Sum Property of a Triangle:**
   In any triangle, the sum of all angles is \( 180^\circ \). Let's denote the angles in the smaller triangle as follows:
   - The vertex angle is \( \alpha \).
   - The base angles are equal, each denoted as \( \theta \).

   Using the angle sum property:
   \[
   2\theta + \alpha = 180^\circ
   \]
   \[
   \theta = \frac{180^\circ - \alpha}{2}
   \]

3. **Relation Between the Two Triangles:**
   The larger triangle shares angle \( \alpha \) at the vertex. Since the triangle is also isosceles, let the base angles of the larger triangle be \( \beta \). So, for the larger triangle:
   \[
   2\beta + \alpha = 180^\circ
   \]
   \[
   \beta = \frac{180^\circ - \alpha}{2}
   \]
   This shows that the angles in the larger triangle are symmetrical to those of the smaller triangle.

4. **Conclusion for \( \angle n \):**
   Since \( \beta \) is the base angle of the large triangle and matches the angle \( n \), we conclude:
   \[
   n = \beta = \frac{180^\circ - \alpha}{2}
   \]

Thus, the angle \( n \) is:

\[
n = \frac{180^\circ - \alpha}{2}
\]

Would you like more details or clarifications on this solution? Here are some related questions to consider:

1. How do isosceles triangles influence the angle properties in geometric problems?
2. What role does symmetry play in solving this type of geometric setup?
3. How can the angle sum property be used in more complex polygons?
4. What would change if the triangles were not isosceles?
5. How could this approach be extended to 3D geometry, such as with pyramids?

**Tip:** Always check for symmetries or equal sides in geometric problems, as they can simplify calculations significantly.

Find angle n in a geometric diagram involving isosceles triangles with given sides and angles.

Solve for angle n in a geometric problem involving two isosceles triangles. Learn how the angle sum property and symmetry play a role in finding the solution. This problem is suitable for students in Grades 9-12 and involves fundamental concepts of triangle geometry.

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