Mathematics
In the figure (i) given below, AB is a diameter of the circle. If ∠ADC = 120°, find ∠CAB.
Circles
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Answer
Join CB as shown in the figure below:
So, ABCD becomes a cyclic quadrilateral.
Sum of opposite angles of cyclic quadrilateral = 180°
⇒ ∠CBA + ∠ADC = 180°
⇒ ∠CBA + 120° = 180°
⇒ ∠CBA = 180° - 120°
⇒ ∠CBA = 60°.
Join AC.
In △ABC
∠ACB = 90° (∵ angle in semicircle is equal to 90°.)
Sum of angles of triangle = 180°
⇒ ∠CAB + ∠ACB + ∠CBA = 180°
⇒ ∠CAB + 90° + 60° = 180°
⇒ ∠CAB + 150° = 180°
⇒ ∠CAB = 180° - 150°
⇒ ∠CAB = 30°.
Hence, the value of ∠CAB = 30°.
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