1.

Theorem : The chords corresponding to congruent arcs of a circle (or congruent circles) are congruent. 

Answer»

Given: B is the centre of circle. arc APC ≅ arc DQE 

To prove: chord AC ≅ chord DE

Proof: 

[m(arc APC) = ∠ABC (i) [Definition of measure of 

m(arc DQE) = ∠DBE] (ii) minor arc] 

arc APC ≅ arc ∠DQE (iii) [Given] 

∴ ∠ABC ≅ ∠DBE [From (i), (ii) and (iii)] 

In ∆ABC and ∆DBE, 

side AB ≅ side DB [Radil of the same circle] 

side [CB] side [EB] [Radii of the same circle] 

∠ABC ≅∠DBE [From (iii), Measures of congruent arcs]

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

∴ chord AC ≅ chord DE [c.s.c.t]


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