Question

10
, ,
10. In the figure, given below, ABCD is a cyclic
quadrilateral in which angleBAD = 75°; angleABD
= 58° and angle ADC = 77º. Find :
(i) BCA
​

Answer 1

AnswEr :

⠀⠀⠀⠀

We're provided with the information that, ABCD is a cyclic quadrilateral.

⠀⠀⠀⠀

• ∠BAD = 75°

⠀⠀⠀⠀

• ∠ABD = 58°

⠀⠀⠀⠀

• ∠ADC = 77°

⠀⠀⠀⠀

We are supposed to find out the ∠BCA !

⠀⠀⠀⠀

Now,

⠀⠀⠀⠀

$\longrightarrow$ ∠ADC + ∠ABD = 180° [Opposite angles are always supplementary ]

⠀⠀⠀⠀

$\longrightarrow$ 77 + ∠ABD = 180

⠀⠀⠀⠀

$\longrightarrow$ ∠ABD = 180 - 77

⠀⠀⠀⠀

$\longrightarrow$ ∠ABD = 103°

⠀⠀⠀⠀

Also, we know that,

$\longrightarrow$ ∠CBD = ∠ABC - ∠ABD [From the diagram]

⠀⠀⠀⠀

$\longrightarrow$ ∠CBD = 103 - 58

⠀⠀⠀⠀

$\longrightarrow$ ∠CBD = 45°

⠀⠀⠀⠀

Since,

⠀⠀⠀⠀

$\longrightarrow$ ∠DAB + ∠BCD = 180 [Opposite angles are always supplementary ]

⠀⠀⠀⠀

$\longrightarrow$ ∠BCD = 180 - ∠DAB

⠀⠀⠀⠀

$\longrightarrow$ ∠BCD = 180 - 75

⠀⠀⠀⠀

$\longrightarrow$ ∠BCD = 105°

⠀⠀⠀⠀

Therefore, ∠BCD = 105°