Let's analyze both the Assertion (A) and the Reason (R) statements to determine the correct option.

Assertion (A):

In triangle $PQR$, $S$ and $T$ are points on $PQ$ and $PR$ respectively. If $\frac{PS}{SQ} = \frac{PT}{TR}$ and $\angle PST = \angle PRQ$, then $\triangle PQR$ is an isosceles triangle.

Reason (R):

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Analysis:

1. Understanding Assertion (A):

   • Given $\frac{PS}{SQ} = \frac{PT}{TR}$ and $\angle PST = \angle PRQ$.
   • If $\angle PST = \angle PRQ$, it implies that $ST$ is parallel to $QR$ (by the Alternate Interior Angles Theorem).
   • Since $ST \parallel QR$, by the Basic Proportionality Theorem (also known as Thales' theorem), $\frac{PS}{SQ} = \frac{PT}{TR}$.
   • However, this condition does not necessarily imply that $\triangle PQR$ is isosceles. The triangles could still be scalene unless there is additional information provided.

2. Understanding Reason (R):

   • Reason (R) correctly states the Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally.

Conclusion:

• The Reason (R) correctly explains a property related to triangles and proportionality.
• However, Reason (R) does not directly justify why $\triangle PQR$ would be an isosceles triangle. The fact that $\frac{PS}{SQ} = \frac{PT}{TR}$ and $\angle PST = \angle PRQ$ holds true for any triangle, not just an isosceles triangle.

Thus, the Assertion (A) is false (since it doesn't necessarily mean $\triangle PQR$ is isosceles), and the Reason (R) is true.

Correct Option:

• (C) A is false, but R is true.

Would you like to dive deeper into this topic, or have any more questions about this problem?

Here are five additional questions you could explore:

1. What is the Basic Proportionality Theorem, and how is it applied in triangle problems?
2. What are the conditions necessary for a triangle to be classified as isosceles?
3. Can a triangle be isosceles if the proportionality condition holds but without the parallel condition?
4. How does the Angle Bisector Theorem relate to the Basic Proportionality Theorem?
5. What are other ways to prove that a triangle is isosceles?

Tip: When dealing with geometric proofs, always ensure that every step logically follows from the previous one and directly connects to the final conclusion.