Question

Figure 5
In the given figure 6, AOB is a diameter and ABCD is a cyclic quadrilateral. If, ADC = 120°,
Then. BAC =?​

Answer 1

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

Step-by-step explanation:

angle ACB= 90० ( angle at semi circle)

ADC +ABC = 180 ( opposite sides of cyclic quadrilateral)

120+ABC = 180

ABC = 60

In triangle ACB,

CAB + ACB+ ABC = 180

CAB +90+60 = 180

CAB + 150= 180

CAB = 30