Multiple choice

ABCD is a quadrilateral inscribed in a circle. Diagonals AC and BD are joined. If $\angle CAD = 60^{\circ}$ and $\angle BDC = 25^{\circ}$. Find $\angle BCD$

  1. $85^{\circ}$
  2. $120^{\circ}$
  3. $115^{\circ}$
  4. $95^{\circ}$
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D Correct answer
Explanation

Angles subtended by the same arc are equal. Angle CBD = angle CAD = 60 degrees. In triangle BCD, the sum of angles is 180 degrees. Angle BCD = 180 - 60 - 25 = 95 degrees.

AI explanation

In a cyclic quadrilateral, angles subtended by the same chord in the same segment are equal, so angle BAC equals angle BDC which is 25 degrees. We then calculate angle BAD as 60 degrees plus 25 degrees to get 85 degrees. Since opposite angles in a cyclic quadrilateral are supplementary, angle BCD equals 180 degrees minus 85 degrees, resulting in 95 degrees.

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