To solve this problem, let's break it down step by step:

Given:

1. $ABCD$ is a square.
2. $\triangle BCF$ and $\triangle CDE$ are equilateral triangles.
3. The goal is to find the angle $\angle EGF$.

Steps:

1. Analyze the square and equilateral triangles

• In a square, all internal angles are $90^\circ$.
• Each side of the square is equal in length, so: $AB = BC = CD = DA.$
• In an equilateral triangle, all internal angles are $60^\circ$, and all sides are equal.

2. Consider points $E$, $F$, and $G$:

• Point $E$ lies on the extension of side $CD$, forming $\triangle CDE$.
• Point $F$ lies on the extension of side $BC$, forming $\triangle BCF$.
• $G$ is the intersection of the diagonals of $\triangle BCF$ and $\triangle CDE$.

3. Interior angle at $\angle EGF$:

To find $\angle EGF$, observe the geometry:

• Since $\triangle BCF$ and $\triangle CDE$ are equilateral, $\angle BCF = 60^\circ$ and $\angle CDE = 60^\circ$.
• The angle at $\angle EGF$ is influenced by the overlapping structure.

By using angle properties in this geometry:

• The sum of angles around point $G$ gives: $\angle EGF = 120^\circ.$

Final Answer:

$\angle EGF = 120^\circ.$

Would you like a detailed breakdown with diagrams, or do you have any specific questions?

Related Questions:

1. How is the center of an equilateral triangle calculated geometrically?
2. How do you calculate angles in overlapping triangles?
3. Can you explain the properties of square diagonals in relation to this problem?
4. What are some practical uses of equilateral triangles in geometry?
5. How can this geometry be visualized in a coordinate plane?

Tip: Understanding properties of equilateral triangles helps solve problems involving symmetry and overlapping angles.