Q. 1: In the given figure, PS/SQ = PT/TR and ∠ PST = ∠ PRQ. Prove that PQR is an isosceles triangle. (isosceles triangle, ratio, angles, proportionality, parallel lines)

Step 1: Establish Parallel Lines using Converse of BPT

We are given that the ratio of the sides $PS$ to $SQ$ is equal to the ratio of the sides $PT$ to $TR$. According to the converse of the Basic Proportionality Theorem (also known as Thales Theorem or Thales Intercept Theorem), if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. Therefore, in $\triangle PQR$, the line segment $ST$ is parallel to $QR$.

Step 2: Identify Corresponding Angles

Since $ST$ is parallel to $QR$, and $PQ$ is a transversal line intersecting these parallel lines, the angle $\angle PST$ and $\angle PQR$ are corresponding angles. As per geometric properties, corresponding angles formed by a transversal intersecting parallel lines are equal.

Step 3: Equate Angles

We are given in the problem statement that $\angle PST = \angle PRQ$. From the previous step, we established that $\angle PST = \angle PQR$. By transitive property of equality, if two angles are equal to the same third angle, then they are equal to each other. Hence, $\angle PQR$ must be equal to $\angle PRQ$.

Step 4: Conclusion about Isosceles Triangle

In $\triangle PQR$, we have proven that the angles opposite to the sides $PR$ and $PQ$ are equal (i.e., $\angle PQR = \angle PRQ$). A fundamental property of triangles states that if two angles in a triangle are equal, then the sides opposite to these angles are also equal in length. Therefore, side $PQ$ is equal to side $PR$.

Step 5: Final Proof

Since two sides of $\triangle PQR$, namely $PQ$ and $PR$, are equal in length, by definition, $\triangle PQR$ is an isosceles triangle. An isosceles triangle is a triangle that has at least two sides of equal length.

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