Question

1. Recall that two circles are congruent if they have the same radii. Prove that equal
chords of congruent circles subtend equal angles at their centres.
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Answer 1

Two circles are said to be congruent if and only if their radii are equal.

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Let AB & CD  are two equal chords of two congruent circles with Centre O and O’. i.e AB= CD.

 

To Prove: ∠AOB = ∠CO'D

Proof:

In ΔAOB and ΔCO'D,

AB = CD ( given)

OA = O'C (Radii of congruent circles)

OB = O'D (Radii of congruent circles)

 ΔAOB ≅ ΔCO'D (SSS congruence rule)

 ∠AOB = ∠CO'D (By CPCT)

Hence, equal chords of congruent circles subtend equal angles at their centres.

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Hope this will help you..

Answer 2

Answer:

• Let us consider a circle with center O and two equal chords of a circle AB and CD.
• We need to prove that ∠AOB=∠COD
• In △ AOB and COD, we have AO=CO (Radius of the circle) BO=DO (Radius of the circle)
• AB=CD (Equal chords)
• By SAS criterion of congruence, we have

△AOB≅△COD

⇒∠AOB=∠COD