Multiple choice

ABCD is a cyclic quadrilateral formed on a circle, which has a diagonal AC. E is tangent to the circle at point C. If (\angle ABC = 110^{o}) and (\angle ACD = 40^{o}), find the angle DCE?

  1. $70^o$
  2. $90^o$
  3. $75^o$
  4. $80^o$
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A Correct answer
Explanation

In a cyclic quadrilateral, opposite angles sum to 180°. Given ∠ABC = 110°, then ∠ADC = 180° - 110° = 70°. In triangle ACD, ∠CAD = 180° - 40° - 70° = 70°. The angle between tangent CE and chord CA equals angle in alternate segment: ∠DCE = ∠CAD = 70°. This is the alternate segment theorem.

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