Corresponding arcs of congruent chords of a circle (or congruent circles) are congruent

Question

Theorem: Corresponding arcs of congruent chords of a circle (or congruent circles) are congruent 

Answer 1

Given: O is the centre of circle, chord PQ = chord RS 

To prove: arc PMQ = arc RNS

Proof: 

In ∆POQ and ∆ROS, 

[side PO ≅ side RO side OQ ≅ side OS] 

[Radii of the same circle] 

chord PQ ≅ chord RS [Given] 

∴ ∆POQ ≅ ∆ROS [SSS test of congruency] 

∴ ∠POQ ≅ ∠ROS (i) [c.a.c.t.] 

m(arc PMQ) = ∠POQ (ii) 

m(arc RNS) = ∠ROS (iii) [Definition of measure of minor arc]

∴ arc PMQ ≅ arc RNS [From (i), (ii) and (iii)]