In a ΔABC, points P, Q and R are taken on AB, BC and CA, respectively, such that BQ = PQ and QC = QR. If ∠BAC = 75^o, what is the measure of ∠PQR (in degrees)?

30

Calculation

∠BAC = 75^o

∠ABC + ∠ACB + ∠BAC = 180^o

∠ABC + ∠ACB + 75^o = 180^o

∠ABC + ∠ACB = 105^o

Let, ∠ABC = ∠PBQ = 70^o and ∠ACB = ∠RCQ = 35^o

So, ∠PQR =180^o - (∠PBQ + ∠RCQ)

⇒ 180^o - [(180^o - 2∠PBQ) + (180^o - 2∠RCQ)]   (as BQ = PQ, QC = QR)

⇒ 180^o - [(180^o - 2 x 70^o) + (180^o - 2 x 35^o)]

⇒ 180^o - (40^o + 110^o)

⇒ 180^o - 150^o

⇒ 30^o

Therefore, ∠PQR is 30^o.