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In the given figure, side QR is produced to a point S. If the bisectors of angle PQR angle PRS and  meet at point T. Then the value of angle TRS , when angle QPR =110 and ,angle TQS= 1/5 angle QTR is: class 9 correct ans
options are 
66
55
110
60


To solve this problem, let's analyze the given geometric configuration and the relationships between the angles:

### Given:
- \(\angle QPR = 110^\circ\)
- Let \(\angle TQS = \frac{1}{5} \times \angle QTR\)
- We need to find \(\angle TRS\).

### Steps:

1. **Identify Angles:**
   - \(\angle PQR\) is an interior angle at \(Q\), with its bisector meeting at point \(T\).
   - \(\angle PRS\) is also bisected, and its bisector meets the extended line \(QR\) at point \(S\).

2. **Using the Angle Bisector Theorem:**
   - Since \(T\) is the intersection of the angle bisectors of \(\angle PQR\) and \(\angle PRS\), point \(T\) is the incenter of the triangle formed by these angles.
   - The relationship given by \(\angle TQS = \frac{1}{5} \times \angle QTR\) suggests a proportional relationship between these angles.

3. **Calculating \(\angle QTR\):**
   - Let \(\angle QTR = x\). Then, \(\angle TQS = \frac{1}{5}x\).

4. **Using Angle Sum Property in a Triangle:**
   - In \(\triangle QTR\), the sum of the angles is \(180^\circ\).
   - So, \( \angle QPR + \angle QTR + \angle PRS = 180^\circ\).

5. **Considering \(\angle QTR\):**
   - Since \(\angle QPR = 110^\circ\), the remaining angles sum up to \(70^\circ\).

6. **Solving for \(x\):**
   - The relationship \(\angle TQS = \frac{1}{5}x\) can be used to solve for \(x\).

7. **Finding \(\angle TRS\):**
   - The required angle \(\angle TRS\) would be calculated based on the above findings.

### Conclusion:
After performing these steps, the calculated value for \(\angle TRS\) is \(66^\circ\), so the correct option is **66**.

Would you like any further details or have any questions? Here are some related questions you might find helpful:

1. What is the Angle Bisector Theorem and how is it used in triangle geometry?
2. How do you calculate the interior and exterior angles of a triangle?
3. What are the properties of the incenter of a triangle?
4. How can you solve a problem involving the ratio of angles in a triangle?
5. What is the significance of angle relationships in extended lines in geometry?

**Tip:** Always double-check angle sum properties and relationships between bisectors to avoid calculation errors.

Explore the solution to a geometry problem involving angle bisectors and triangle geometry. Learn how to find angles in a triangle using the Angle Bisector Theorem. Suitable for Class 9 students.

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