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$PQ$ is a tangent touching a circle with centre $O$ and $A$ is the point of contact. If the measure of $∠ PAB = 67 °$ , then the measure of $∠ AQB$ will be

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73 °

64 °

53 °

44 °

answer is D.

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Detailed Solution

Given that

PQ

is a tangent and

∠ P A B = 67 °

.
The chord is a diameter as it is passing through O which is the centre.

∴ ∠ B A C = 90 °

(Angle in a semicircle is

90 °

)
By using the alternate segment theorem, we get,

∠ P A B = ∠ A C B = 67 °

Δ A B C

∠ A B C + ∠ A C B + ∠ B A C = 180 °

⇒ ∠ A B C + 67 ° + 90 ° = 180 °

⇒ ∠ A B C = 23 °

Calculating the value of

∠ B A Q

Δ A B Q

∠ A B Q + ∠ A Q B + ∠ B A Q = 180 °

⇒ 23 ° + ∠ A Q B + 113 ° = 180 °

⇒ ∠ A Q B = 44 °

Therefore, the measure of

∠ A Q B = 44 °

.
Therefore the correct option will be 4.

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