https://file.aiquickdraw.com/mathbot/file/1725981590008orlz3cx3.jpg

In the provided diagram, we are tasked with finding the value of angle ∠CBF. Here are the important elements and observations:

1. **Tangent Line**: \( DE \) is tangent to the circle at point \( B \).
2. **Angles in the Circle**:
   - ∠CAB = 61°
   - ∠CBF = the angle to be found.
   - ∠ACB = 41°

### Steps to solve for ∠CBF:
1. **Use the Tangent-Secant Angle Theorem**:
   - The angle between a tangent and a chord (∠CBF) is equal to the angle subtended by the chord (in this case, chord \( AC \)) at the opposite side of the circle. This is known as the *Alternate Segment Theorem*.
   - ∠CBF = ∠CAB (the angle subtended by the chord \( AC \) at the opposite arc).

Thus, based on the diagram and theorem, we can conclude:

\[
\boxed{∠CBF = 61^\circ}
\]

Let me know if you need further clarification or more details.

Here are some related questions:
1. What is the Alternate Segment Theorem, and how does it apply to tangents and circles?
2. How can we use the properties of cyclic quadrilaterals to solve similar problems?
3. What is the significance of the tangent-secant angle theorem in geometry?
4. Can you explain how angles subtended by the same arc in a circle are equal?
5. What is the role of tangent lines in circle theorems?

**Tip**: Always look for circle theorems (like angles subtended by the same arc) when solving geometry problems involving circles.

Find the value of ∠CBF in the given circle diagram where DE is a tangent to the circle at B, ∠CAB = 61° and ∠ACB = 41°.

Learn how to find the value of angle CBF in a circle geometry problem, using the Alternate Segment Theorem. This problem involves a tangent to the circle at point B and key angles subtended by the chord AC. Suitable for high school students.

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