# In the given figure, PQR is an equilateral triangle and QRST is a square. Prove that(i) PT=PS, (ii) ∠PSR=15°. 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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