Question

In the adjoining 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

Given: O is the centre of the circle, seg RS

is the diameter of the circle. 

To prove: ∠RTS is an acute angle. 

Construction: Let seg RT intersect the circle at point P. Join PS and PT.

Proof: 

seg RS is the diameter. [Given] 

∴ ∠RPS = 90° [Angle inscribed in a semicircle] 

Now, ∠RPS is the exterior angle of ∆PTS. 

∴ ∠RPS > ∠PTS [Exterior angle is greater than the remote interior angles]

∴ 90° > ∠PTS 

i.e. ∠PTS < 90° 

i.e, ∠RTS < 90° [R – P -T] 

∠RTS is an acute angle.