Question

A quadrilateral ABCD is inscribed in a circle such that AB is a diameter and ∠ADC = 130º. Find ∠BAC.

Answer 1

Draw a quadrilateral ABCD inscribed in a circle having centre O.
 
Given, ∠ADC = 130°

Since, ABCD is a quadrilateral inscribed in a circle, therefore ABCD becomes a cyclic quadrilateral.

∵ Since, the sum of opposite angles of a cyclic quadrilateral is 180°.

∴ ∠ADC + ∠ABC = 180°

⇒ 130° + ∠ABC = 180°

⇒ ∠ABC = 50°

Since, AB is a diameter of a circle, then AB subtends an angle to the circle is right angle.

∴ ∠ACB = 90°

In ΔABC, ∠BAC + ∠ACB + ∠ABC = 180°  ...[By angle sum property of a triangle]

⇒ ∠BAC + 90° + 50° = 180°

⇒ ∠BAC = 180° – (90° + 50°)

= 180° – 140°

= 40°