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In this problem, we are tasked with finding the size of angle \( ABC \). Here's the step-by-step solution:

1. **Given Information:**
   - Triangle \( DEC \) is equilateral, meaning all its internal angles are \( 60^\circ \).
   - Angle \( \angle DCE = 60^\circ \).
   - Angle \( \angle EAC = 80^\circ \).

2. **Understanding the Geometry:**
   - \( \triangle ABC \) is a larger triangle that contains \( \triangle DEC \).
   - We need to find \( \angle ABC \) (the angle at vertex \( B \)).

3. **Using Angle Sum in Triangle \( AEC \):**
   - In triangle \( AEC \), we know:
     - \( \angle EAC = 80^\circ \).
     - \( \angle DEC = 60^\circ \) because triangle \( DEC \) is equilateral.
   - The sum of the internal angles in any triangle is \( 180^\circ \).
   \[
   \angle AEC = 180^\circ - \angle EAC - \angle DEC
   \]
   \[
   \angle AEC = 180^\circ - 80^\circ - 60^\circ = 40^\circ
   \]

4. **Using Exterior Angle Theorem:**
   - Now, observe that angle \( ABC \) is an exterior angle to triangle \( AEC \).
   - The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two opposite interior angles.
   - Therefore, \( \angle ABC = \angle EAC + \angle AEC \):
   \[
   \angle ABC = 80^\circ + 40^\circ = 120^\circ
   \]

Thus, the size of angle \( ABC \) is \( 120^\circ \).

Would you like more details or have any questions? 

Here are 5 related questions:
1. How do you calculate the angles of an equilateral triangle?
2. What is the exterior angle theorem and how is it applied?
3. How can the sum of angles in a triangle help solve more complex geometry problems?
4. How does symmetry play a role in understanding equilateral triangles?
5. How can other properties of triangles (isosceles, scalene) affect angle calculations?

**Tip:** When solving for angles, always consider using the exterior angle theorem, as it provides a direct relationship between internal and external angles in a triangle.

The shaded triangle DEC is an equilateral triangle. What is the size of angle ABC?

Solve the geometry problem of finding angle ABC where the shaded triangle DEC is equilateral. Learn about the exterior angle theorem and angle sum property in triangles. Suitable for Grades 8-10.

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