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)
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Step-by-Step Solution
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 △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 ∠PST and ∠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 ∠PST=∠PRQ. From the previous step, we established that ∠PST=∠PQR. By transitive property of equality, if two angles are equal to the same third angle, then they are equal to each other. Hence, ∠PQR must be equal to ∠PRQ.
Step 4: Conclusion about Isosceles Triangle
In △PQR, we have proven that the angles opposite to the sides PR and PQ are equal (i.e., ∠PQR=∠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 △PQR, namely PQ and PR, are equal in length, by definition, △PQR is an isosceles triangle. An isosceles triangle is a triangle that has at least two sides of equal length.