Prove that if chords of congruent circles subtend equal angles at their centres,

Question. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.


Solution:

Let us consider two congruent circles (circles of same radius) with centres as O and O'.



In $\triangle \mathrm{AOB}$ and $\triangle C O^{\prime} D$,

$\angle \mathrm{AOB}=\angle \mathrm{CO}^{\prime} \mathrm{D}($ Given $)$

$O A=O^{\prime} C$ (Radii of congruent circles)

$O B=O^{\prime} D$ (Radii of congruent circles)

$\therefore \triangle \mathrm{AOB} \cong \triangle \mathrm{CO}^{\prime} \mathrm{D}(\mathrm{SAS}$ congruence rule $)$

$\Rightarrow A B=C D(B y C P C T)$

Hence, if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

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