Question

In the adjoining figure, AB is the diameter of the circle with centre O. Two tangents p and g are drawn to the circle at points A and B respectively. Prove that p || q. Further, a line CD touches the circle at E and ∠BCD = 110°. Find the measure of ∠ADC.

Answer 1

Given that, O is centre of circle and p, q are tangents at point A and B.

As we know, tangent to a circle is perpendicular to the radius at the point of contact.

OB ⊥ q and OA ⊥ p

⇒ ∠OBC = 90° and ∠OAD = 90°

Since, ∠ABC + ∠BAD = 90° + 90°

= 180°

⇒ Tangents p and q are parallel to each other.

∵ CD is tangent as well as transversed line.

∴ By co-interior angles,

∠ADC + ∠BCD = 180°

∠ADC + 110° = 180°

∴ ∠ADC = 70°

Or 

∠D = 70°