MathematicsState true or false.In given figure if AOB is a diameter and ∠ADC = 120°, then ∠CAB = 30°.

State true or false.


In given figure if AOB is a diameter and ∠ADC = 120°, then ∠CAB = 30°.


  1. A
    True
  2. B
    False 

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    Solution:

    Given that, AOB   is a diameter and ∠ADC= 120 °   .
    From the figure, we can say that ABCD   is a cyclic quadrilateral and thus the sum of either pair of opposite angles is 180 °  .
    So, ∠ADC+∠ABC= 180 °   .
    Since ∠ADC= 120 °   , so 120 ° +∠ABC= 180 °   .
    ⇒∠ABC= 180 ° − 120 °   ⇒∠ABC= 60 °   Considering triangle ABC  ,
    ∠ACB= 90 °   (Angle in a semicircle is 90 °  )
    Applying the sum of angles in a triangle is 180 °  :
    ∠ABC+∠ACB+∠CAB= 180 °   ∠ABC+∠ACB+∠CAB= 180 °   60 ° + 90 ° +∠CAB= 180 °   ∠CAB= 180 ° − 150 °   ∠CAB= 30 °  
    Thus, the given statement is true.
    Hence, option 1 is correct.
     
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