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Let's analyze the given problem step by step:

### Key Information:
1. \( EF \) and \( FG \) are tangents to the circle at points \( E \) and \( G \), respectively.
2. We need to find the size of \( \angle EFG \).
3. The following theorems are provided as hints:
   - The angle subtended at the center of a circle is twice the angle subtended at the circumference for the same arc.
   - A tangent is perpendicular to the radius at the point of contact.

### Steps to Solve:
1. **Tangent-Radius Theorem**: Since \( EF \) is a tangent to the circle at \( E \) and \( FG \) is a tangent to the circle at \( G \), the angles between the tangents and the radii \( OE \) and \( OG \) are both \( 90^\circ \). This means:
   \[
   \angle OEF = 90^\circ \quad \text{and} \quad \angle OGF = 90^\circ
   \]
   
2.Continuing from the tangent-radius property, we can proceed to use the information provided.

2. **Angle at the Center**:
   We are given \( \angle DOG = 52^\circ \). According to the circle theorem, the angle subtended at the center of a circle is **twice** the angle subtended at the circumference for the same arc. Therefore, the angle \( \angle EFG \), subtended by the same arc \( DG \) at the point \( F \) on the circumference, is half of \( \angle DOG \):
   \[
   \angle EFG = \frac{1}{2} \times \angle DOG = \frac{1}{2} \times 52^\circ = 26^\circ
   \]

Thus, the size of \( \angle EFG \) is \( 26^\circ \).

### Final Answer:
\[
\boxed{\angle EFG = 26^\circ}
\]

Let me know if you'd like more details or have further questions.

Here are five questions to expand this understanding:

1. How can we prove that a tangent is always perpendicular to the radius at the point of contact?
2. Can the tangent-radius property be extended to non-circular curves like ellipses?
3. How would the solution change if \( \angle DOG \) were different?
4. What other circle theorems are useful when dealing with angles formed by tangents?
5. What would \( \angle DGF \) be if \( \angle DOG \) were instead \( 104^\circ \)?

**Tip**: Always remember to use the key circle theorems involving tangents and radii, as they simplify many geometry problems.

In the diagram, EF and FG are tangents to the circle at E and G respectively. Find the size of ∠EFG.

Learn how to solve a geometry problem involving tangents and circle theorems to find the angle EFG. This problem applies key theorems such as the tangent-radius theorem and the angle subtended at the center theorem.

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