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In the second theorem, are the major arcs corresponding to congruent chords congruent?

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Yes, the major arcs corresponding to congruent chords are congruent.

Proof: 

In ∆POQ and ∆ROS, 

seg OP ≅ seg OR [Radii of the same circle] 

seg OQ ≅ seg OS [Radii of the same circle] 

seg PQ ≅ seg RS [Given] 

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

∴ ∠POQ ≅ ∠SOR (i) [c.a.c.t]

[ m(arc PMQ) = ∠POQ (ii) 

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

∴ m(arc PMQ) = m(arc RNS) 

m(minor arc) = 360° – m(major arc) (iv) [From (i), (ii) and (iii)] 

m(arc PMQ) = 360° – m(arc PXQ) (v) 

and m(arc RNS) = 360° – m(arc RXS) (vi) 

∴ 360°- m(arc PXQ) = 360°- m(arc RXS) [From (iv), (v) and (vi)] 

∴ m(arc PXQ) = m(arc RXS)

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