Question

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

Answer 1

Given: $B$ is the centre of circle.
$arcAPC\cong \mathrm{arc}DQE$
To prove: chord $AC\cong$ chord $DE$
$\left[m\right(arcAPC)=\angle ABC$ (i) [Definition of measure of
$m\left(\mathrm{arc}DQE\right)=\angle DBE]$ (ii) minor arc]
$arcAPC\cong \mathrm{arc}\angle DQE$ (iii) [Given]
$\therefore \angle ABC\cong \angle DBE[$ From (i), (ii) and (iii)]
In $△ABC$ and $△DBE$,
side $AB\cong$ side $DB$ [Radil of the same circle]
side [CB] side [EB] [Radii of the same circle]
$\angle ABC\cong \angle DBE$ [From (iii), Measures of congruent arcs]
$\therefore △ABC\cong △DBE[SAS$ test of congruency]
$\therefore$ chord $AC\cong$ chord $DE$ [c.s.c.t]