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ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see Fig. 7.34). Show that BCD is a right angle.Q.6

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Its very important question of chapter triangles ncert class 9th . Find the best and simplest solution of exercise 7.2 question 6 ,Please help me to solve this question in a best and simplest way.ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see Fig. 7.34). Show that BCD is a right angle.

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  1. It is given that AB = AC and AD = AB

    We will have to now prove BCD is a right angle.

    Proof:

    Consider ΔABC,

    AB = AC (It is given in the question)

    Also, ACB = ABC (They are angles opposite to the equal sides and so, they are equal)

    Now, consider ΔACD,

    AD = AB

    Also, ADC = ACD (They are angles opposite to the equal sides and so, they are equal)

    Now,

    In ΔABC,

    CAB + ACB + ABC = 180°

    So, CAB + 2ACB = 180°

    ⇒ CAB = 180° – 2ACB — (i)

    Similarly, in ΔADC,

    CAD = 180° – 2ACD — (ii)

    also,

    CAB + CAD = 180° (BD is a straight line.)

    Adding (i) and (ii) we get,

    CAB + CAD = 180° – 2ACB+180° – 2ACD

    ⇒ 180° = 360° – 2ACB-2ACD

    ⇒ 2(ACB+ACD) = 180°

    ⇒ BCD = 90°

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