Question

Prove that two arcs of circle are congruent if angles subtended by them at center arc equal.

Answer 1

Given : A circle with center $O$ and two angles formed at center.
$\angle AOB=\angle COD$ ……(i)
To Prove :$\widehat{AB}=\widehat{CD}$
Construction : Join $OA,OB,OC$ and $OD$ by center $O$.
Proof : In $\Delta AOB$ and $\Delta COD$ (given)
$AO=OD$ (radius of circle)
$OB=OC$ (radius of circle)
$\angle AOB=\angle COD$ (given)
By SAS congruence criterion
$\Delta AOB=\Delta COD$
Thus, By CPCT
$AB=CD$
Chord $AB$ = Chord $CD$ and $AB=CD$