This is circle based question from Chapter name- circles

Topic – Angle properties of circles

Chapter number- 15

In this question we have been given the figure of circle with certain information about the angles and we have to find the value of x.

ICSE Avichal publication

Understanding ICSE Mathematics

Question no 1, exercise 15.2

Solution:From the figure

(i) ABCD is a cyclic quadrilateral

Solution:From the figure

(i) ABCD is a cyclic quadrilateral

Ext. ∠DCE = ∠BAD

∠BAD = x

^{o}Now arc BD subtends ∠BOD at the center

And ∠BAD at the remaining part of the circle.

∠BOD = 2 ∠BAD = 2 x

2 x = 150

^{o }(x = 75^{0})(ii) ∠BCD + ∠DCE = 180

^{o}(Linear pair)

∠BCD + 80

^{0 }= 180^{o}∠BCD = 180

^{0 }– 80^{0 }= 100^{o}Arc BAD subtends reflex ∠BOD at the

Centre and ∠BCD at the remaining part of the circle

Reflex ∠BOD = 2 ∠BCD

X

^{o }= 2 × 100^{o }= 200^{o}(iii) In ∆ACB,

∠CAB + ∠ABC + ∠ACB = 180

^{o}(Angles of a triangle)

But

∠ACB = 90

^{o}((Angles of a semicircle)

25

^{o }+ 90^{o }+ ∠ABC = 180^{o}=115

^{o }+ ∠ABC = 180^{o}∠ABC = 180

^{o }– 115^{0}=65^{o}ABCD is a cyclic quadrilateral

∠ABC + ∠ADC = 180

^{o}(Opposite angles of a cyclic quadrilateral)

65

^{o }+ x^{o }=180^{o}x

^{o}= 180^{o}-65^{o}= 115^{o}