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. âˆ DCE = âˆ BAD

âˆ BAD = x

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

And âˆ BAD at the remaining part of the circle.

âˆ BOD = 2 âˆ BAD = 2 x

2 x = 150

^{oÂ }(x = 75^{0})(ii) âˆ BCD + âˆ DCE = 180

^{o}(Linear pair)

âˆ BCD + 80

^{0Â }= 180^{o}âˆ BCD = 180

^{0Â }â€“ 80^{0Â }= 100^{o}Arc BAD subtends reflex âˆ BOD at the

Centre and âˆ BCD at the remaining part of the circle

Reflex âˆ BOD = 2 âˆ BCD

X

^{oÂ }= 2 Ã— 100^{oÂ }= 200^{o}(iii) In âˆ†ACB,

âˆ CAB + âˆ ABC + âˆ ACB = 180

^{o}(Angles of a triangle)

But

âˆ ACB = 90

^{o}((Angles of a semicircle)

25

^{oÂ }+ 90^{oÂ }+ âˆ ABC = 180^{o}=115

^{oÂ }+ âˆ ABC = 180^{o}âˆ ABC = 180

^{oÂ }â€“ 115^{0}=65^{o}ABCD is a cyclic quadrilateral

âˆ ABC + âˆ ADC = 180

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

65

^{oÂ }+ x^{oÂ }=180^{o}x

^{o}Â = 180^{o}Â -65^{o}Â = 115^{o}