Q. 65.0( 1 Vote )

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

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.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Basic Proportionality Theorem42 mins
Quiz | Criterion of Similarity of Triangle45 mins
Champ Quiz | Thales Theorem49 mins
How to Ace Maths in NTSE 2020?36 mins
NCERT | Strong Your Basics of Triangles39 mins
RD Sharma | Imp. Qs From Triangles41 mins
R.D Sharma | Solve Exercise -4.2 and 4.3FREE Class
R.D Sharma | Solve Exercise-4.545 mins
NCERT | Basic Proportionality Theorem22 mins
RD Sharma | Imp Qs Discussion- Triangles43 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses