Question

In figure, if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and ∠BQR = 70°, then ∠AQB is equal to ______.

 

Options

• 20°

• 40°

• 35°

• 45°

Answer 1

In figure, if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and ∠BQR = 70°, then ∠AQB is equal to 40°.

Explanation:

Given, AB ॥ PR

∴ ∠ABQ = ∠BQR = 70°  ...[Alternate angles]

Also, QD is perpendicular to AB and QD bisects AB.

In ∆QDA and ∆QDB,

∠QDA = ∠QDB  ...[90° each]

AD = BD

QD = QD   ...[Common side]

∆ADQ ≅ ∆BDQ   ...[By SAS congruency]

⇒ ∠QAD = ∠QBD  [CPCT]...(i)

But ∠QBD = ∠ABQ = 70°

⇒ ∠QAD = 70°   ...[From (i)]

Now, in ∆ABQ,

∠A + ∠B + ∠AQB = 180°   ...[Angle sum property]

⇒ ∠AQB = 180° – (70° + 70°) = 40°