Q. 84.7( 124 Votes )

# ABCD is a cyclic

Answer :

We know that the sum of the measures of opposite angles in a cyclic quadrilateral is 180°. Therefore, ∠ A + ∠ C = 180

⇒4*y* + 20 - 4*x* = 180

⇒- 4*x* + 4*y* = 160

*⇒x* - *y* = - 40 ....(*i*)

Also, ∠ B + ∠ D = 180

⇒3*y* - 5 - 7*x* + 5 = 180

⇒- 7*x*+ 3*y* = 180 .... (*ii*)

Multiplying equation (*i*) by 3, we obtain 3*x* - 3*y* = - 120.... (*iii*)

Adding equations (*ii*) and (*iii*), we obtain

- 7*x* + 3*x* = 180 - 120

- 4*x* = 60

*x* = -15

By using equation (*i*), we obtain *x* - *y* = - 40

-15 - *y* = - 40

*y* = -15 + 40

= 25

∠ A = 4*y* + 20

= 4(25) + 20 **∠A= 120°**

∠ B = 3*y* - 5

= 3(25) - 5 **∠B= 70°**

∠C = - 4*x* = - 4(- 15)

**∠C = 60°**

∠D = - 7x + 5

= - 7(-15) + 5

= 105 + 5

** ∠D = 110°**

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