Q. 224.3( 6 Votes )

In the given figure, PQR is an equilateral triangle and QRST is a square. Prove that

(i) PT=PS, (ii) PSR=15°.


Answer :

Given: PQR is an equilateral triangle and QRST is a square


To prove: PT=PS and PSR=15°.


Proof:


Since ∆PQR is equilateral triangle,


PQR = PRQ = 60°


Since QRTS is a square,


RQT = QRS = 90°


In ∆PQT,


PQT = PQR + RQT


= 60° + 90°


= 150°


In ∆PRS,


PRS = PRQ + QRS


= 60° + 90°


= 150°


PQT = PRS


Thus in ∆PQT and ∆PRS,


PQ = PR … sides of equilateral triangle


PQT = PRS


QT = RS … side of square


Thus by SAS property of congruence,


∆PQT PRS


Hence, we know that, corresponding parts of the congruent triangles are equal


PT = PS


Now in ∆PRS, we have,


PR = RS


PRS = PSR


But PRS = 150°


SO, by angle sum property,


PRS + PSR + SPR = 180°


150° + PSR + SPR = 180°


2PSR = 180° - 150°


2PSR = 30°


PSR = 15°


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