Question

While proving the first theorem of the two, we assume that the minor arc APC and minor arc DQE are congruent. Can you prove the same theorem by assuming that corresponding major arcs congruent?

Answer 1

Statement: 

The chords corresponding to congruent major arcs of a circle are congruent. 

Given: B is the centre of circle. arc AXC ≅ arc DXE 

To prove: chord AC ≅ chord DE

Proof: 

m(major arc) = 360° – m(minor arc) 

∴ m(arc AXC) = 360° – m(arc APC) (i)

m(arc DXE) = 360° – m(arc DQE) (ii) 

m(arc AXC) = m(arc DXE) (iii) [Given] 

∴ 360° – m(arc APC) = 360°- m(arc DQE) 

[From (i), (ii) and (iii)] 

∴ m(arc APC) = m(arc DQE) (iv) 

∴ m(arc APC) = ∠ABC (v) [Definition of measure of minor arc] 

m(arc DQE) = ∠DBE (vi) 

∴ ∠ABC = ∠DBE (vii) [From (iv), (v) and (vi)] 

In ∆ABC and ∆DBE, 

[side AB ≅ side DB Side CB ≅ side EB] [Radii of the same circle]

∠ABC ≅ ∠DBE [From (vii)] 

∴ ∆ABC ≅ ∆DBE [SAS test of congruency]

∴ chord AC ≅ chord DE [c.s.c.t.]