In figure if ∠AOB=90o and ∠ABC=30o. Then, which of the following is the measure of ∠CAO?

Consider the given figure,

From the figure-

AB

is the chord of a circle with centre

O

∠ AOB = 90^{o}

∠ ABC = 30^{o}

The angle subtended by a chord to the circumference of a circle is half of that subtended to the centre.
Here,

AB

subtends

∠ AOB

to the centre and

∠ ACB

to the circumference.
Then,

∠ ACB = \frac{1}{2} ∠ AOB

∠ ACB = \frac{1}{2} \times 90^{o}

∠ ACB = 45^{o}

In triangle

ABC,

by angle sum property,

∠ BAC + ∠ ACB + ∠ ABC = 180^{o}

∠ BAC = 180^{o} - (∠ ACB + ∠ ABC)

∠ BAC = 180^{o} - (45^{o} + 30^{o})

∠ BAC = 180^{o} - 75^{o}

∠ BAC = 105^{o}

In triangle

AOB

AO = BO

[Radius]
Hence, triangle

AOB

is an isosceles triangle.
In an isosceles triangle angle opposite to equal sides are equal.
Then,

∠ OAB = ∠ OBA

.
By angle sum property,

⟹ ∠ AOB + ∠ OAB + ∠ OBA = 180^{o}

⟹ ∠ AOB + 2 ∠ OAB = 180^{o}

{⟹ 90}^{o} + 2 ∠ OAB = 180^{o}

⟹ 2 ∠ OAB = 180^{o} - 90^{o}

⟹ {2 ∠ OAB = 90}^{o}

⟹ ∠ OAB = 45^{o}

Then,

⟹ ∠ CAO = ∠ BAC - ∠ OAB

⟹ ∠ CAO = 105^{o} - 45^{o}

⟹ ∠ CAO = 60^{o}

Hence, the measure of

∠ CAO

60^{o}

.
Therefore, option 2 is correct.

In figure if $∠ AOB = 90^{o} and ∠ ABC = 30^{o} .$ Then, which of the following is the measure of $∠ CAO$ ?