Question

In a pentagon PQRST, ∠P = 100°, ∠Q = 120° and ∠S = ∠T. The sides PQ and SR, when produced meet at right angle. Find ∠QRS and ∠PTS.

Answer 1

In the figure, PQ and SR produced meet at point P,
∴ ∠U = 90°
∠Q = 120°
⇒∠UQR = 180° - 120° = 60°
∴ ∠URQ = 90° - ∠UQR = 90° - 60° = 30°
∴ QRS = 180° - ∠URQ = 180° - 30° = 150°
Let ∠S = ∠T = x
Since, ∠P + ∠Q + ∠QRS + ∠S + ∠T = (5 - 2) x 180°
⇒100° + 120° + 150° + x + x = 3 x 180°
⇒370° + 2x = 540°
⇒ 2x = 170°
⇒ x = 85°
HEnce, ∠PTS = 85°.