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Question 1. If O is the centre of the circle, find the value of x in each of the following figures (using the given information):

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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

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2 Answers

  1. Solution:

    From the figure

    (i) ABCD is a cyclic quadrilateral

    ML Aggarwal Solutions for Class 10 Chapter 15 - Image 16

    Ext. ∠DCE = ∠BAD

    ∠BAD = xo

    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(x = 750)

    (ii) ∠BCD + ∠DCE = 180o

    (Linear pair)

    ML Aggarwal Solutions for Class 10 Chapter 15 - Image 17

    ∠BCD + 80= 180o

    ∠BCD = 180– 80= 100o

    Arc BAD subtends reflex ∠BOD at the

    Centre and ∠BCD at the remaining part of the circle

    Reflex ∠BOD = 2 ∠BCD

    X= 2 × 100= 200o

    (iii) In ∆ACB,

    ML Aggarwal Solutions for Class 10 Chapter 15 - Image 18

    ∠CAB + ∠ABC + ∠ACB = 180o

    (Angles of a triangle)

    But
    ∠ACB = 90o

    ((Angles of a semicircle)

    25+ 90+ ∠ABC = 180o

    =115+ ∠ABC = 180o

    ∠ABC = 180– 1150=65o

    ABCD is a cyclic quadrilateral

    ∠ABC + ∠ADC = 180o

    (Opposite angles of a cyclic quadrilateral)

    65+ x=180o

    xo = 180o -65o = 115o

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