The chord AB of a circle of centre O subtends an angle Θ with tangent at A to the circle. ∠ABO is?
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Θ
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90° - Θ
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90° + Θ
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2(π - Θ)
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None of these
By the alternate segment theorem, the angle between the tangent and the chord equals the angle in the alternate segment. If the angle between tangent and chord AB is theta, then the angle subtended by chord AB at the circumference is theta. The angle at the center is 2*theta. In triangle OAB, OA=OB (radii), so angle OAB = angle OBA = (180 - 2*theta) / 2 = 90 - theta.
The tangent at point A is perpendicular to the radius OA, making angle OAB plus the given angle theta equal to 90 degrees. Because OA and OB are radii of the same circle, triangle OAB is an isosceles triangle, meaning angle OAB equals angle ABO. Substituting angle ABO for angle OAB in our first equation gives angle ABO plus theta equals 90 degrees. Therefore, angle ABO equals 90 degrees minus theta.