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.


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

Thus, PMRS is a parallelogram.


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))


In ∆SPQ and ∆RQP,


PQ QP (same line segment)

SP RQ (given)

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


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

In ∆TPS and ∆QTR

STP RTQ (Vertically opposite angles)


∆TPS and ∆QTR are similar by AA corollary.

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

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