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Question
In the following figure, if AOB is a diameter and ∠ADC = 120°, then ∠CAB = 30°.
Options
True
False
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Solution
This statement is True.
Explanation:
Let AOB be the diameter of the circle.
Given: ∠ADC = 120°
Firstly, join CB.
Then, we have a cyclic quadrilateral ABCD.
Since sum of opposite angles of cyclic quadrilateral is 180°, therefore
∠ADC + ∠ABC = 180°
⇒ 120° + ∠ABC = 180°
⇒ ∠ABC = 180° – 120°
⇒ ∠ABC = 60°
Now join AC.
Also, diameter subtends a right angle to the circle,
∴ In ΔABC, ∠ACB = 90°
Now, by angle sum property of a triangle, sum of all angles of a triangle is 180°.
∴ ∠CAB + ∠ABC + ∠ACB = 180°
⇒ ∠CAB + 60° + 90° = 180°
⇒ ∠CAB = 180° – 90° – 60°
⇒ ∠CAB = 30°
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