In the adjoining diagram, PQ is a tangent at A to the circle with centre O. If ∠OAC = 25°, then ∠ABC is:

1. 20°

2. 65°

3. 70°

4. 130°

Given,

∠OAC = 25°

We know that,

Radius from the center and tangent are perpendicular to each other at point of contact.

⇒ ∠OAQ = ∠OAP = 90°

From figure,

⇒ ∠OAQ = ∠OAC + ∠CAQ

⇒ 90° = 25° + ∠CAQ

⇒ ∠CAQ = 90° - 25°

⇒ ∠CAQ = 65°.

We know that,

The angle between a tangent and a chord through the point of contact is equal to an angle in alternate segment. Hence,

⇒ ∠ABC = ∠CAQ

⇒ ∠ABC = 65°.

Hence, option 2 is the correct option.