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.
Solution:
Given:
Two Congruent Circles C1 and C2
AB is the chord of C1
and PQ is the chord of C2
AB = PQ
To Prove: Angle subtended by the Chords AB and PQ are equal i.e. ∠AOB = ∠PXQ
Proof:
In △AOB & △PXQ
AO = PX (Radius of congruent circles are equal)
BO = QX (Radius of congruent circles are equal)
AB = PQ (Given)
△AOB ⩭ △PXQ (SSS congruence rule)
Therefore, ∠AOB = ∠PXQ (CPCT)
Question 2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
Solution:
Given:
Two Congruent circles C1 and C2
AB is the chord of C1 and PQ is chord of C2
& ∠AOB = ∠PXQ
To prove :
In △AOB and △PXQ ,
AO = PX (Radius of congruent circles are equal)
∠AOB = ∠PXQ (Given)
BO = QX (Radius of congruent circles are equal)
△AOB ⩭ △PXQ (SAS congruence rule)
Therefore, AB = PQ (CPCT)