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.