Question

ABCD is a cyclic quadrilateral in which:

(i) BC || AD, ∠ADC = 110° and ∠BAC = 50°. Find ∠DAC.

(ii) ∠DBC = 80° and ∠BAC = 40°. Find ∠BCD.

(iii) ∠BCD = 100° and ∠ABD = 70° find ∠ADB.

Answer 1

(i) It is given that , and

We have to find

In cyclic quadrilateral ABCD

..… (1)

..… (2)

Since,

So,

Therefore in ,

So , ..… (3)

Now, ( and is transversal)

(ii) It is given that , and

We have to find

_{(Angle in the same segment are equal)}

_Hence,

(iii) It is given that, ∠BCD = 100° and ∠ABD = 70°

As we know that sum of the opposite pair of angles of cyclic quadrilateral is 180°.

$$\begin{gathered} \angle DAB+\angle BCD=180° \\ \Rightarrow \angle DAB=180°-100° \\ =80° \end{gathered}$$

In ΔABD we have,

$$\begin{gathered} \angle DAB+\angle ABD+\angle BDA=180° \\ \Rightarrow \angle BDA=180°-150°=30° \end{gathered}$$

_Hence,