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
In $△POQ$ and $△ROS$,
[side $PO\cong$ side $RO$
side $OQ\cong$ side $OS$ ] [Radii of the same circle]
chord $PQ\cong$ chord $RS$ [Given]
$\therefore △POQ\cong △ROS$ [SSS test of congruency]
$\therefore \angle POQ\cong \angle ROS$ (i) [c.a.c.t.]
$m\left(\mathrm{arc}PMQ\right)=\angle POQ$ (ii)
$m\left(\mathrm{arc}RNS\right)=\angle ROS$ (iii) [Definition of measure of minor arc]
$\therefore \mathrm{arc}PMQ\cong arcRNS[$ From (i), (ii) and (iii)]