CDS 2026 (I) Elementary Mathematics, question 46

Topic: Geometry

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Q46

In a cyclic quadrilateral ABCD, the diagonal AC bisects the angle C. Which of the following statements is/are correct ?

I. ∠ABD=∠ADB\angle ABD=\angle ADB

II. The diagonal BD is parallel to the tangent of the circle at A

Select the answer using the code given below :

  1. (a) I only
  2. (b) II only
  3. (c) Both I and IICorrect answer
  4. (d) Neither I nor II

Solution.

Answer: (c) Both I and II

  1. Since ∠BCA=∠ACD\angle BCA = \angle ACD, the intercepted chords BA and AD are equal.
  2. Thus triangle ABD is isosceles, giving ∠ABD=∠ADB\angle ABD = \angle ADB: I is true.
  3. By the tangent–chord theorem, the angle between the tangent at A and AB equals ∠ADB\angle ADB.
  4. This equals the angle between DB and BA, establishing that the tangent is parallel to BD.