if indeed the QT || RS, that simply means that QT is the midsegment of ∡PRS, but that also means that the corresponding sides of the small triangle ∡PQT and the larger one ∡PRS are in a proportion, meaning
[tex] \bf \cfrac{PQ}{PR}=\cfrac{PT}{PS}~\hspace{5em}\cfrac{18}{18+38}=\cfrac{36}{36+80}\implies \cfrac{18}{56}=\cfrac{36}{116}\implies \cfrac{9}{28}\ne \cfrac{9}{29}~~\bigotimes [/tex]
