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In the given diagram, we are tasked with finding the size of \( \angle EFG \). The hints provided suggest the use of the following circle theorems:

1. **The angle subtended at the center is twice the angle subtended at the circumference for the same arc.**
2. **A tangent is perpendicular to the radius.**

### Analyzing the Diagram:
- \( EF \) and \( FG \) are tangents to the circle at points \( E \) and \( G \) respectively.
- \( \angle DGO = 52^\circ \), which is subtended by arc \( DE \).
- \( \angle EOF \) is subtended at the center by the same arc \( DE \).

### Step-by-step Solution:
1. **Angle at the center vs angle at the circumference:**
   By the first circle theorem, \( \angle EOF = 2 \times \angle DGO = 2 \times 52^\circ = 104^\circ \).

2. **Isosceles triangle:**
   \( OE \) and \( OG \) are radii of the circle, forming an isosceles triangle \( OEG \). The angle at the center, \( \angle EOF \), forms the vertex angle of this isosceles triangle. Since the sum of the angles in a triangle is \( 180^\circ \), the base angles are calculated as:
   \[
   \text{Base angle} = \frac{180^\circ - \angle EOF}{2} = \frac{180^\circ - 104^\circ}{2} = \frac{76^\circ}{2} = 38^\circ.
   \]

3. **Angle \( \angle EFG \):**
   \( \angle EFG \) is an exterior angle of triangle \( OEG \), and by the exterior angle theorem, it equals the sum of the opposite interior angles \( \angle OEG \) and \( \angle OGE \). Since both these angles are \( 38^\circ \), we get:
   \[
   \angle EFG = 38^\circ + 38^\circ = 76^\circ.
   \]

Thus, the size of \( \angle EFG \) is \( \boxed{76^\circ} \).

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Would you like more details or explanations on any step?

Here are 5 related questions to explore:
1. How does the exterior angle theorem apply to other triangle problems?
2. Why is the angle subtended at the center always twice the angle subtended at the circumference?
3. What other circle theorems could be applied to this type of problem?
4. How does the property of tangents being perpendicular to the radius affect other angles in the diagram?
5. Can you find the length of the sides if the radius of the circle was given?

**Tip:** Always label known angles clearly in circle geometry diagrams to avoid confusion when applying theorems.

In the diagram, EF and FG are tangents to the circle at E and G respectively. Find the size of ∠EFG. (Hint: Use the following circle geometry theorems)

Learn how to find the size of angle ∠EFG in a circle with tangents EF and FG using circle geometry theorems. This problem covers essential concepts such as the exterior angle theorem and the angle subtended by an arc theorem. Suitable for students in Grades 9-11.

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