Question

ABCD is a cyclic quadrilateral in a circle with centre O. If ∠ADC = 130°; find ∠BAC.

In the given figure, ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle through A, B, C, D. If ∠ADC =130°, find ∠BAC.

ABCD is a cyclic quadrilateral with O as centre of the circle. If ∠ADC = 130°; find ∠BAC.

Answer 1

We know that,

Angle in a semi-circle is 90°

∠ACB = 90°

We know that,

Sum of opposite angles of a cyclic quadrilateral = 180°

⇒ ∠ABC = 180° − ∠ADC = 180° − 130° = 50°

In △ACB,

⇒ ∠ACB + ∠CBA + ∠BAC = 180° [Angle sum property]

⇒ 90° + 50° + ∠BAC = 180°

⇒ ∠BAC + 140° = 180°

⇒ ∠BAC = 180° − 140° = 40°

Hence, ∠BAC = 40°