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

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Solution

Given:
AOB is a diameter of circle
∠ADC = 120°

Quadrilateral ADCB is a cyclic quadrilateral.

In a cyclic quadrilateral, the sum of opposite angles is 180^∘.

Thus, ∠ADC + ∠CBA = 180°
⇒ 120° + ∠CBA = 180°
⇒ ∠CBA = 180° − 120°
⇒ ∠CBA = 60° ...(1)

We know, the diameter subtends a right angle to any point on the circle.
∴ ∠ACB = 90° ...(2)

In ∆ACB,
∠CAB + ∠CBA + ∠ACB = 180° (angle sum property)
⇒ ∠CAB + 60° + 90° = 180° (From (1) and (2))
⇒ ∠CAB = 180° − 150°
⇒ ∠CAB = 30° ...(3)

Hence, ∠CAB = 30°.

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