Multiple choice

$\Box ABCD$ is inscribed in a circle such that $AB$ is a diameter and $ \angle ADC={ 130 }^{ o }$, then $m\angle BAC= $

  1. $ { 90 }^{ o } $
  2. $ { 50 }^{ o } $
  3. $ { 40 }^{ o } $
  4. $ { 30 }^{ o } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In a cyclic quadrilateral, opposite angles sum to 180. Angle ABC + Angle ADC = 180, so Angle ABC = 50. Since AB is a diameter, Angle ACB = 90. In triangle ABC, Angle BAC = 180 - 90 - 50 = 40 degrees.

AI explanation

Because quadrilateral ABCD is cyclic, the opposite angles must sum to 180 degrees by the properties of cyclic quadrilaterals. Given angle ADC is 130 degrees, angle ABC equals 180 - 130 = 50 degrees. Since AB is the diameter, angle ACB is 90 degrees by the Thales theorem. In triangle ABC, the sum of angles is 180 degrees, so angle BAC is 180 - 90 - 50 = 40 degrees. The correct option is C.

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