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The problem in the image is about proving that the triangle \( \triangle PQR \) is an isosceles triangle using given conditions and properties of parallel lines and angles. Let's go through the proof step-by-step.

### Given:
1. \( \frac{PS}{SQ} = \frac{PT}{TR} \)
2. \( \angle PST = \angle PRQ \)

### To Prove:
\( \triangle PQR \) is an isosceles triangle.

### Proof:
1. From the given ratio \( \frac{PS}{SQ} = \frac{PT}{TR} \), by the converse of the Basic Proportionality Theorem (Theorem 6.2), line \( ST \parallel QR \).

2. Since \( ST \parallel QR \) and \( \angle PST = \angle PQR \) are corresponding angles, they are equal.

3. In triangle \( \triangle PQR \), we have:
   - \( \angle PST = \angle PQR \)

4. Since two angles in a triangle are equal, \( \triangle PQR \) is an **isosceles triangle** by the property that a triangle with two equal angles has two equal sides.

Thus, we have proved that \( \triangle PQR \) is an isosceles triangle.

### Would you like further details or have any questions?

### Related Questions:
1. What is the Basic Proportionality Theorem (Thales' Theorem) and how is it used?
2. How can we prove that a triangle is isosceles using sides instead of angles?
3. What are the properties of parallel lines used in triangle proofs?
4. What is the converse of the Basic Proportionality Theorem?
5. How can the concept of similar triangles help in proving triangle properties?

### Tip:
In geometry, using properties of parallel lines and angles often provides a straightforward way to prove the equality of angles and sides in triangles.

In Fig. 6.16, PS/SQ = PT/TR and ∠PST = ∠PRQ. Prove that PQR is an isosceles triangle.

Learn how to prove that triangle PQR is an isosceles triangle using the Basic Proportionality Theorem and corresponding angles in geometry. This solution covers key concepts of parallel lines and the converse of Thales' Theorem, suitable for Grades 9-10.

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