Question

prove that "Chords of a circle which subtends equal angles at the centre are equal​

Answer 1

Step-by-step explanation:

Given :-

The angles subtended by two chords at the centre are equal.

Required To Prove:-

Chords are equal.

Proof :-

O is the centre of the circle .

PQ and RS are the two chords of the circle

∠POQ and ∠ROS are subtended by the two chords at the centre of the circle .

Given that

Two chords PQ and RS of a circle subtend equal angles at centre of a circle.

∠POQ ≅∠ROS

In ∆ POQ and ∆ ROS,

OP ≅ OR (radius )

∠POQ ≅∠ROS (Given )

OQ ≅ OS (Radius )

By SAS theorem of congruence

∆ POQ ≅ ∆ ROS

=> PQ ≅ RS

Since , Corresponding parts are congruent in congruent triangles

Hence, Chords are equal.

Hence, Proved.

Used Theorem :-

SAS Property:-

If in two triangles, Two sides and the included angle in the first triangle are congruent to the corresponding two sides and the included angle in the other triangle ,then the two triangles are congruent.