example related to converse of Basic Proportionality Theorem or Thales Theorem

Example 1. In a figure,PS/SQ =PT/TR and ∠PST = ∠PRQ. prove that PQR is an isosceles triangle.
Solution:- it is given that PS/SQ =PT/TR

           so,   ST∥QR  ( converse of Basic Proportionality Theorem)
Therefore,  ∠PST = ∠PQR (corresponding angles)........(1)
Also,       it is given that 
                   ∠PST =∠PRQ...............(2)
so,              ∠ PRQ= ∠PQR [ from (1) and (2)]
Therefore,        PQ = PR (sides opposite the equal angles)
i.e.,  PQR is an isosceles triangle.
                          
                               


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