Question

Fill In The Blanks

ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscibing it and ∠ADC = 140^∘ ,then ∠ BAC = ________.

Answer 1

Given:
ABCD is a cyclic quadrilateral
AB is a diameter of the circle circumscribing ABCD
∠ADC = 140^∘


In a cyclic quadrilateral, the sum of opposite angles is 180^∘.


Thus, ∠ADC + ∠ABC = 180°
⇒ 140° + ∠ABC = 180°
⇒ ∠ABC = 180° − 140°
⇒ ∠ABC = 40° ...(1)

Since AB is the diameter of the circle
Therefore, ∠ACB = 90° (angle in the semi circle) ...(2)

In ∆ABC,
∠BAC + ∠ACB + ∠ABC = 180° (angle sum property)
⇒ ∠BAC + 90° + 40° = 180° (From (1) and (2))
⇒ ∠BAC + 130° = 180°
⇒ ∠BAC = 180° − 130°
⇒ ∠BAC = 50°


Hence, ∠BAC = 50°.