Question

In the given figure, seg RS is a diameter of the circle with centre O. Point T lies in the exterior of the circle. Prove that ∠ RTS is an acute angle.

Answer 1

Let seg RT intersect circle at point M. Draw seg MS.

Angle inscribed in a semicircle is a right angle.

∴ ∠RMS = 90°

∴ seg SM ⊥ side RT

∴ ΔTMS is a right angled triangle.

∴ ∠TMS + ∠MTS + ∠TSM = 180°   ...(Sum of all angles of a triangle is 180°)

∴ 90° + ∠MTS + ∠TSM = 180°

∴ ∠MTS + ∠TSM = 180° − 90°

∴ ∠MTS + ∠TSM = 90°

∴ ∠MTS < 90°

i.e., ∠RTS  < 90°   ...(R-M-T)

∴ ∠RTS is an acute angle.   ...(By definition)