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°

2∠PSR = 180° - 150°

2∠PSR = 30°

∠PSR = 15°