In figure, PQRS is a square and SRT is an equilateral triangle. Prove

1.	In figure, PQRS is a square and SRT is an equilateral triangle. Prove that (i) PT = QT (ii) ∠ TQR = 15°
Answer» Given: PQRS is a square and SRT is an equilateral triangle. To prove: (i) PT =QT and (ii) ∠TQR = 15° Now, PQRS is a square: PQ = QR = RS = SP …… (i) And ∠ SPQ = ∠ PQR = ∠ QRS = ∠ RSP = 90° Also, △ SRT is an equilateral triangle: SR = RT = TS …….(ii) And ∠ TSR = ∠ SRT = ∠ RTS = 60° From (i) and (ii) PQ = QR = SP = SR = RT = TS ……(iii) From figure, ∠TSP = ∠TSR + ∠ RSP = 60° + 90° = 150° and ∠TRQ = ∠TRS + ∠ SRQ = 60° + 90° = 150° ∠ TSR = ∠ TRQ = 150° ………………… (iv) By SAS congruence criterion, Δ TSP ≃ Δ TRQ We know, corresponding parts of congruent triangles are equal So, PT = QT Proved part (i). Now, consider ΔTQR. QR = TR [From (iii)] Δ TQR is an isosceles triangle. ∠QTR = ∠TQR [angles opposite to equal sides] Sum of angles in a triangle = 180° ∠QTR + ∠ TQR + ∠TRQ = 180° 2∠TQR + 150° = 180° [From (iv)] 2 ∠TQR = 30° ∠TQR = 15° Hence proved part (ii).

In figure, PQRS is a square and SRT is an equilateral triangle. Prove that (i) PT = QT (ii) ∠ TQR = 15°

Answer»

Given: PQRS is a square and SRT is an equilateral triangle.

To prove: (i) PT =QT and (ii) ∠TQR = 15°

Now,

PQRS is a square:

PQ = QR = RS = SP …… (i)

And ∠ SPQ = ∠ PQR = ∠ QRS = ∠ RSP = 90°

Also, △ SRT is an equilateral triangle:

SR = RT = TS …….(ii)

And ∠ TSR = ∠ SRT = ∠ RTS = 60°

From (i) and (ii)

PQ = QR = SP = SR = RT = TS ……(iii)

From figure,

∠TSP = ∠TSR + ∠ RSP

= 60° + 90°

= 150° and

∠TRQ = ∠TRS + ∠ SRQ

= 60° + 90°

= 150°

∠ TSR = ∠ TRQ = 150° ………………… (iv)

By SAS congruence criterion, Δ TSP ≃ Δ TRQ

We know, corresponding parts of congruent triangles are equal

So, PT = QT

Proved part (i).

Now, consider ΔTQR.

QR = TR [From (iii)]

Δ TQR is an isosceles triangle.

∠QTR = ∠TQR [angles opposite to equal sides]

Sum of angles in a triangle = 180°

∠QTR + ∠ TQR + ∠TRQ = 180°

2∠TQR + 150° = 180° [From (iv)]

2 ∠TQR = 30°

∠TQR = 15°

Hence proved part (ii).