Question

In Fig. 10.23, PQRS is a square and SRT is an equilateral triangle. Prove that

Answer 1

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Q. PQRS is a square and SRT is an equilateral triangle. prove that PT=QT and TQR = 15 degree.

Ans. Since PQRS is a square,

angle PSR = angle QRS... (each 90°)

Now, again in equilateral triangle SRT,

we have angle TSR = angle TRS ...(each 60°)

Angle PSR + Angle TSR = Angle QRS + Angle TRS

⇒ Angle TSP = Angle TRQ

Now in Δ TSP and Δ TRQ, we have

TS = TR (Sides of equilateral triangle)

Angle TSP = Angle TRQ... (Proved above)

PS = QR ...(Sides of square)

Δ TSP ≡ Δ TRQ

So, PT = QT Proved.

Now in Δ TQR, we have

TR = QR ....(QR = RS = TR)

Angle TQR = Angle QTR and Angle TQR + Angle QTR + Angle TRQ = 180°

Angle TQR + Angle QTR + Angle TRS + Angle SRQ = 180°

2(Angle TQR) + 60° + 90° = 180° .....(∴ ∠ TQR = ∠ QTR)

2(Angle TQR) = 180° - 150°

2(Angle TQR) = 30°

Angle TQR = 30/2

Angle TQR = 15°

Hence proved

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