https://file.aiquickdraw.com/mathbot/file/1726370148231rxd492ro.jpg

To solve the problem in the image, let's use the given information step by step:

### Given:
- The diagram represents a circle with center \( O \).
- \( EF \) and \( FG \) are tangents to the circle at points \( E \) and \( G \), respectively.
- \( \angle DOE = 52^\circ \) (angle subtended at the center of the circle).
- You need to find the size of \( \angle EFG \).

### Using Circle Geometry Theorems:
1. **The angle subtended at the center is twice the angle subtended at the circumference on the same arc.**
   - In this case, \( \angle DOE = 52^\circ \) is subtended at the center, which means the angle subtended at the circumference on the same arc \( DE \) is half of this.
   - Therefore, \( \angle DGE = \frac{52^\circ}{2} = 26^\circ \).

2. **Tangents to a circle form a right angle (90°) with the radius at the point of contact.**
   - \( EF \) and \( FG \) are tangents, so \( \angle OEF = 90^\circ \) and \( \angle OGF = 90^\circ \).

3. **Angle at a tangent point:**
   - \( \angle EFG \) is an external angle in the triangle formed by the tangents. This angle is equal to the sum of the opposite internal angles, which are \( \angle OEF \) and \( \angle OGF \).
   - So, \( \angle EFG = 90^\circ + 26^\circ = 116^\circ \).

### Answer:
The size of \( \angle EFG = 116^\circ \).

Would you like more details on this solution or have any questions? Here are 5 related questions you can explore:

1. How do we derive the relationship between angles subtended at the center and the circumference?
2. Why are angles formed by a tangent and a radius always 90°?
3. What is the significance of an external angle in a cyclic quadrilateral?
4. Can tangents from a point outside a circle ever intersect at more than one point on the circle?
5. How does the property of tangents help in proving the congruence of triangles formed by tangents?

**Tip:** Always remember the key circle theorem that says "the angle subtended at the center of a circle is twice the angle subtended at the circumference." This is crucial in many geometry problems!

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

Solve a circle geometry problem involving tangents to a circle. Find the size of ∠EFG using key theorems, such as the angle subtended at the center being twice that subtended at the circumference. Suitable for Grades 10-12.

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