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Question

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°

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Solution

The correct option is (b).
In a cyclic quadrilateral ABCD:
A=x+y+10°
B=y+20°
C=x+y-30°
D=x+y°
We have:
A+C=180° and B+D=180° [Since ABCD is a cyclic quadrilateral]
Now, A+C=x+y+10°+x+y-30°=180°
⇒ 2x + 2y − 20 = 180
⇒ x + y − 10 = 90
⇒ x + y = 100 ....(i)
Also, B+D=y+20°+x+y°=180°
⇒ x + 2y + 20 = 180
⇒ x + 2y = 160 ....(ii)
On subtracting (i) from (ii), we get:
y = (160 − 100) = 60
On substituting y = 60 in (i), we get:
x + 60 = 100 ⇒ x = (100 − 60) = 40
∴​ B=y+20°=60+20°=80°

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