Q. 214.2( 5 Votes )

# In a cyclic quadrilateral ABCD, it is being given that ∠A = (x + y + 10)°, ∠B = (y + 20)°, ∠C = (x + y – 30)° and ∠D = (x + y)°. Then, ∠B = ?

A. 70°

B. 80°

C. 100°

D. 110°

Answer :

It is given in the question that,

In cyclic quadrilateral ABCD, we have:

∠ A = (x + y + 10)^{o}

∠ B = (y + 20)^{o}

∠ C = (x + y – 30)^{o}

∠ D = (x + y)^{o}

As ABCD is a cyclic quadrilateral

∴ ∠ A + ∠ C = 180^{o} and ∠ B + ∠ D = 180^{o}

Now, ∠ A + ∠ C = 180^{o}

(x + y + 10)^{o} + (x + y – 30)^{o} = 180^{o}

2x + 2y – 20^{o} = 180^{o}

x + y = 100^{o} (i)

Also, ∠ B + ∠ D = 180^{o}

(y + 20)^{o} + (x + y)^{o} = 180^{o}

x + 2y + 20^{o} = 180^{o}

x + 2y = 160^{o} (ii)

On subtracting (i) from (ii), we get

y = (160 – 100)^{o}

y = 60^{o}

Putting the value of y in (i), we get

x + 60^{o} = 100^{o}

x = 100^{o} – 60^{o}

x = 40^{o}

∴ ∠ B = (y + 20)^{o}

∠ B = 60^{o} + 20^{o} = 80^{o}

Hence, option B is correct

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