In the given diagram, chord AB = chord BC.
(i) what is the relation between arcs AB and BC?
(ii) what is the relation between ∠AOB and ∠BOC?
(iii) If arc AD is greater than arc ABC, then what is the relation between chords AD and AC?
(iv) If ∠AOB = 50°, find the measure of angle BAC.
Join OA, OB, OC and OD.
(i) Arc AB = Arc BC [ ∵ Equal chords subtends equal arcs]
(ii) ∠AOB = ∠BOC [ ∵ Equal chords subtends equal arcs]
(iii) If arc AD > arc ABC, then chord AD > AC
(iv) ∠AOB = 50°
But ∠AOB = ∠BOC [from (ii) above]
∴ ∠BOC = 50°
Now arc BC subtends ∠BOC at the center and ∠BAC at The remaining part of the circle.
∴ `∠BAC = 1/2 ∠BOC = 1/2xx 50° = 25°`