Multiple choice

A quadrilateral ABCD is inscribed in a circle with centre O. If ∠BOC = 92° and ∠ADC = 112°, then ∠ABO is equal to:

  1. 46o

  2. 48o

  3. 24o

  4. 56o
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C Correct answer
Explanation

∠BOC = 92° is central angle subtended by arc BC. ∠ADC = 112° is inscribed angle subtended by arc ABC (opposite arc). Central angle for arc ABC = 2 × 112° = 224°. Total central angles = 360°, so central angle for arc ADC = 360° - 224° = 136°. ∠ABD subtends arc AD = 136°/2 = 68°. In triangle ABD: 68° + 112° + ∠ABO = 180°, so ∠ABO = 24° (or use: central angle for arc BDC is 92° + 68° = 160°, inscribed angle BAC = 80°, then triangle ABO gives ∠ABO = 180° - 92° - 64° = 24°).

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