In Fig. 6.16, SQPS=TRPT and ∠PST=∠PRQ. Prove that PQR is an isosceles triangle. [Figure: triangle PQR with vertex P at top; points S on PQ and T on PR with segment ST drawn; angle PST is marked at S.]
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Step-by-Step Solution
Step 1: State the given information
We are given two conditions. The first condition describes the ratio of the segments on sides PQ and PR, and the second condition states that angle PST is equal to angle PRQ.
Step 2: Apply the converse of Basic Proportionality Theorem
According to the converse of the Basic Proportionality Theorem (also known as Thales Theorem or Thales's Theorem), if a line divides two sides of a triangle proportionally, then the line is parallel to the third side. Here, line segment ST divides sides PQ and PR of triangle PQR proportionally, so ST must be parallel to QR.
Step 3: Identify corresponding angles
When two parallel lines (ST and QR) are intersected by a transversal line (PQ), the corresponding angles formed are equal. Therefore, angle PST is equal to angle PQR.
Step 4: Use the given angle equality
We are given that ∠PST=∠PRQ. From the previous step, we established that ∠PST=∠PQR. By substituting ∠PST with ∠PQR in the given condition, we conclude that ∠PQR=∠PRQ.
Step 5: Conclude that triangle PQR is isosceles
In any triangle, if two angles are equal, then the sides opposite to these equal angles are also equal. Since ∠PQR=∠PRQ in △PQR, the side opposite to ∠PQR (which is PR) must be equal to the side opposite to ∠PRQ (which is PQ). This means △PQR is an isosceles triangle with PQ = PR.