Question

prove that two chords of a circle are congruent if and only if they subtend arcs of equal measure.

Answer 1

Let the arc AB be equal to the arc SK. It is required to prove that the chord AB is equal to the chord CK. We connect the ends of the chords with the center of the circle - point O. The resulting triangles AOB and KOS are equal, since they have two, respectively, equal sides (radii of one circle) and at an equal angle between these sides (these angles are equal, as central, corresponding to equal arcs ). Therefore, AB = CK.