Multiple choice

A quadrilateral ABCD is inscribed in a circle so that AB is the diameter of the circle. If $\angle$ ADC = 115$^{\circ}$, find $\angle$ BAC.

  1. 20$^{\circ}$
  2. 25$^{\circ}$
  3. 30$^{\circ}$
  4. 40$^{\circ}$
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B Correct answer
Explanation

In a cyclic quadrilateral, opposite angles sum to 180 degrees. If angle ADC = 115, then angle ABC = 180 - 115 = 65 degrees. Since AB is the diameter, angle ACB = 90 degrees (angle in a semicircle). In triangle ABC, angle BAC = 180 - 90 - 65 = 25 degrees.

AI explanation

Because ABCD is a cyclic quadrilateral, opposite angles must sum to 180 degrees. With angle ADC at 115 degrees, angle ABC equals 180 minus 115, which is 65 degrees. Since AB is the diameter, the angle subtended at the circumference, angle ACB, is 90 degrees. In triangle ABC, the angles add to 180 degrees, so angle BAC is 180 minus 90 minus 65, which equals 25 degrees.

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