Question

In the figure, O is the centre of the circle and ∠AOB = 90°, ∠ABC = 30°. Then find ∠CAB.

Answer 1

Given: ∠AOB = 90°, ∠ABC = 30°

To find: ∠CAB

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°