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To find angle \( \angle ABC \), we need to analyze the geometry of the triangle using the information provided in the diagram. Let's break it down:

### Information from the diagram:
1. \( \triangle ABC \) is a triangle with:
   - \( \angle A = 60^\circ \)
   - \( \angle AHE = 80^\circ \)
   
2. Line segment \( BD \) and \( CE \) seem to be equal in length, as indicated by the identical markings, suggesting \( BD = CE \). 

### Step-by-step process to find \( \angle ABC \):

1. **Triangle \( \triangle AHE \):**
   - Since \( \angle AHE = 80^\circ \) and the sum of angles in a triangle is always \( 180^\circ \), we can calculate \( \angle HEA \) and \( \angle HAE \) if more data were available on their relationships (like parallel lines or equal angles). 
   
2. **Isosceles triangle \( \triangle ABC \):**
   - Given that \( BD = CE \), triangle \( \triangle ABC \) appears to be isosceles. In an isosceles triangle, two base angles are equal.
   - Thus, \( \angle ABC = \angle ACB \).

3. **Using the sum of angles in a triangle:**
   - The sum of the angles in any triangle is \( 180^\circ \).
   - Let \( \angle ABC = \angle ACB = x \). Then, we have:
   \[
   60^\circ + x + x = 180^\circ
   \]
   Simplifying the equation:
   \[
   60^\circ + 2x = 180^\circ
   \]
   \[
   2x = 120^\circ
   \]
   \[
   x = 60^\circ
   \]
   
Thus, \( \angle ABC = 60^\circ \).

Would you like further details on any part of the solution? Here are some related questions:

1. What is the significance of isosceles triangles in geometry?
2. How do you calculate the sum of the angles in a polygon?
3. What is the relationship between angle bisectors and isosceles triangles?
4. Can we apply the Law of Sines to verify this result?
5. How would this change if additional angles or side lengths were given?

**Tip:** Always check if a triangle has equal sides or angles, as this often simplifies calculations significantly.

Learn how to calculate angle ABC in an isosceles triangle using the sum of angles and triangle properties. Includes a detailed breakdown of key concepts such as the Isosceles Triangle Theorem and basic angle relationships.

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