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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°
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