Question

AOB is a diameter and ABCD is a cyclic quadrilateral.If ADC =120° .Find BAC​

Answer 1

Step-by-step explanation:

Required Answer:-

We know that, the angle subtend by the diameter or semi-circle at any point of the circle is 90°.

Then:

In the above circle, AB is a diameter because O is the centre. Then, ∠ACB = 90°.

Now:

Another property of cyclic quadrilaterals says that, the opposite angles add upto 180°. That means,

∠CDB + ∠CBA = 180°

∠BCD + ∠DAB = 180°

Considering the first equation, We have ∠CDB

⇒ 120° + ∠CBA = 180°

⇒ ∠CBA = 60°

We have got two out of three angles in ∆CBA, and the third angle is ∠BAC, which we have to find. By angle sum property of triangles::

⇒ ∠ABC + ∠BCA + ∠BAC = 180°

⇒ 60° + 90° + ∠BAC = 180°

⇒ ∠BAC + 150° = 180°

⇒ ∠BAC = 30°

Therefore:

The required unknown angle ∠BAC is 30°.

Answer 2

Answer:

(b) 30°

We have:

∠ABC + ∠ADC = 180° (Opposite angles of a cyclic quadrilateral)

⇒ ∠ABC + 120° = 180°

⇒ ∠ABC = (180° - 120°) = 60°

Also, ∠ACB = 90° (Angle in a semicircle)

In ΔABC, we have:

∠BAC + ∠ACB + ∠ABC = 180° (Angle sum property of a triangle)

⇒ ∠BAC + 90° + 60° = 180°

⇒ ∠BAC = (180° - 150°) = 30°

Step-by-step explanation:

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