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To solve for the angle \( e \), let's analyze the given circle and the geometry of the figure.

### Observations:
1. **Tangents and Angles**:  
   - \( RQ \) and \( TP \) are tangents to the circle at points \( R \) and \( T \), respectively.
   - By the tangent-secant theorem, the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Thus, \( \angle TRS = \angle RTP \).

2. **Circle Properties**:  
   - The sum of the opposite angles of a quadrilateral inscribed in a circle is \( 180^\circ \).

3. **Given Angles**:
   - \( \angle RST = e^\circ \) (let's solve for this).
   - \( \angle TRP = 105^\circ \).

### Step-by-Step Solution:

1. **Finding the Relationship Between Angles**:
   - Since \( RQ \) and \( TP \) are tangents, the angles they form with their respective chords are equal to the angles in the alternate segments.
   - Therefore:
     \[
     \angle RST = \angle RTP = 105^\circ
     \]
   - Therefore, \( e = 105^\circ \).

### Conclusion:
The value of angle \( e \) is \( \boxed{105^\circ} \).

Would you like further details or explanations? Here are some related questions you might be interested in:

1. How do you find the angles in a cyclic quadrilateral?
2. What is the tangent-secant theorem, and how is it applied in problems involving circles?
3. How do you determine the angles formed by tangents and chords in a circle?
4. Why does the sum of opposite angles in a cyclic quadrilateral equal \(180^\circ\)?
5. What are some other properties of circles and tangents that could be useful in geometry?

**Tip:** Remember that angles in the same segment of a circle are always equal.

Learn how to solve for angle e in a circle geometry problem involving tangents RQ and TP. This problem explores the tangent-secant theorem and angles in a cyclic quadrilateral. Suitable for high school students studying circle geometry.

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