MathematicsIn the given figure, which of the following is the value of a?

In the given figure, which of the following is the value of a?


  1. A
    30∘
  2. B
    40∘
  3. C
    60∘
  4. D
    90∘  

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    Solution:

    Consider the given figure,
    From the figure-
    AOB is the diameter of the given circle.
    ABCD is a cyclic quadrilateral.
    AC is joined.
    Also,
    ∠ADC=130∘.
    Since, AOB is the diameter of the given circle, it subtends ∠ACB to the circumference at C, i.e. ∠ACB=90∘...[ since it is an angle in a semicircle].
    Sum of opposite angles of a cyclic quadrilateral is supplementary.
    Then,
    ∠ADC+∠ABC=180∘
    ⇒∠ABC=180∘-∠ADC ⇒∠ABC=180∘-130∘ ⇒∠ABC=50∘ In △ABC, by angle sum property,
    ∠ABC+∠ACB+∠CAB=180∘ ∠CAB=180∘-(∠ACB+∠ABC) ∠CAB=180∘-(90∘+50∘) ∠CAB=a=40∘ Hence, the value of a is 40∘.
    Therefore, option 2 is correct.
     
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