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Question
In the figure, O is the centre of the circle, and ∠AOB = 90°, ∠ABC = 30°. Then find ∠CAB.
Solution
Given: ∠AOB = 90°, ∠ABC = 30°
To find: ∠CAB
Solution:
In given figure,
∠AOB = m (arc AB) ......[Definition of measure of minor arc]
∴ m(arc AB) = 90° ......(i)
Also, ∠ACB = `1/2` m(arc AB) .....[Inscribed angle theorem]
∴ ∠ACB = `1/2 xx 90^circ` ......[From (i)]
∴ ∠ACB = 45° .....(ii)
In ∆ACB,
∠CAB + ∠ABC + ∠ACB = 180° ......`[("Sum of the measures of"),("angles of a triangle is" 180^circ)]`
∴ ∠CAB + 30° + 45° = 180° ......[From (ii)]
∴ ∠CAB + 75° = 180°
∴ ∠CAB = 180° – 75°
∠CAB = 105°
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