Q. 65.0( 1 Vote )

In PQRS, = {T}, PS = QR,. Prove that ΔTPS is similar to ΔQTR.

Answer :


Given: In PQRS, PR QS = {T}, PS = QR and PQ||RS.


To prove: ∆TPS and ∆QTR are similar.


Construction: Through R, draw a line parallel to PS to intersect PQ at M.


Proof:


In PMRS, PM||RS (P-M-Q) and PS||MR (construction).


Thus, PMRS is a parallelogram.


PS = MR


Further, PS = QR (given)


In ∆RMQ, RMQ RQM …(i)


Further, the corresponding angles formed by transversal PQ to PS||MR are congruent.


RMQ SPM (corresponding angles)


RQM SPM (by (i))


RQP SPQ (P-M-Q)


In ∆SPQ and ∆RQP,


SPQ RQP


PQ QP (same line segment)


SP RQ (given)


The correspondence ∆SPQ∆RQP is a congruence by SAS theorem of congruence.


PSQ QRP


PST QRT (S-T-Q and R-T-P)


In ∆TPS and ∆QTR


STP RTQ (Vertically opposite angles)


PST QRT


∆TPS and ∆QTR are similar by AA corollary.


Thus, by AA corollary the correspondence ∆TSP∆TRQ is a similarity.


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