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In the given figure (not drawn to scale), if PS = SR = QR and angle QRT= 87 , then angle PQR  is: class 9 correct ans 
87
56
58
43.5


Given the geometry of the figure:

1. \( PS = SR = RQ \), meaning that \( \triangle PSR \) and \( \triangle QSR \) are both isosceles triangles.

2. The angle \( \angle QRT = 87^\circ \).

We are asked to find \( \angle PQR \).

### Steps to solve:

1. Since \( PS = SR \), \( \triangle PSR \) is isosceles, implying \( \angle SPR = \angle PRS = x \).

2. Similarly, since \( SR = RQ \), \( \triangle QSR \) is isosceles, implying \( \angle SQR = \angle QSR = y \).

3. In \( \triangle PSR \), the sum of angles is \( \angle PSR + \angle SPR + \angle PRS = 180^\circ \). Therefore,
   \[
   \angle PSR + 2x = 180^\circ \quad \text{or} \quad \angle PSR = 180^\circ - 2x.
   \]

4. Similarly, in \( \triangle QSR \), 
   \[
   \angle QSR + 2y = 180^\circ \quad \text{or} \quad \angle QSR = 180^\circ - 2y.
   \]

5. Since \( PS = SR = RQ \), \( \triangle PSR \) and \( \triangle QSR \) are congruent, so \( x = y \).

6. Now consider \( \angle QRT \). Since the exterior angle of a triangle is equal to the sum of the two opposite interior angles:
   \[
   \angle QRT = \angle PQR + \angle PRQ = 87^\circ.
   \]

Since \( \angle PQR \) and \( \angle PRQ \) are equal (from isosceles \( \triangle PQR \)), we can let \( \angle PQR = \angle PRQ = \theta \).

Thus, 
\[
2\theta = 87^\circ \quad \Rightarrow \quad \theta = \frac{87^\circ}{2} = 43.5^\circ.
\]

So, \( \angle PQR \) is **43.5°**.

### Correct Answer:
The correct answer is **43.5°**.

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Would you like further details or have any other questions?

Here are some related questions you might be interested in:
1. What is the sum of angles in an isosceles triangle?
2. How do you calculate the exterior angle of a triangle?
3. What are the properties of an isosceles triangle?
4. How do you prove that two triangles are congruent?
5. What is the relationship between interior and exterior angles in polygons?

**Tip:** Remember that in any triangle, the exterior angle is always equal to the sum of the two non-adjacent interior angles.

Learn how to calculate angles in isosceles triangles with a detailed solution. Given PS = SR = RQ and angle QRT = 87°, find angle PQR. Explore the properties of isosceles triangles and the application of the Exterior Angle Theorem. Suitable for Class 9 students.

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