Two tangents PA and PB are drawn from an external point P to a circle with centre O, such that ∠ = ∠ and ∠ = . Prove that opposite angles are supplementary.

Book Online Demo

Check Your IQ

Try Test

Courses

Offline Centres

Two tangents PA and PB are drawn from an external point P to a circle with centre O, such that ∠𝐴𝑃𝐵 = ∠𝑥 and ∠𝐴𝑂𝐵 = 𝑦. Prove that opposite angles are supplementary.

see full answer

Talk to JEE/NEET 2025 Toppers - Learn What Actually Works!

Real Strategies. Real People. Real Success Stories - Just 1 call away

An Intiative by Sri Chaitanya

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

It is given that Two tangents 𝑃𝐴 and 𝑃𝐵 are drawn from an external point 𝑃 to a circle with centre O, such that ∠𝐴𝑃𝐵and ∠𝐴𝑂𝐵 = 𝑦

We know that the tangent at a point to a circle is perpendicular to radius through that point. Therefore 𝑂𝐴⟂𝐴𝑃 and 𝑂𝐵⟂𝐵𝑃

So ∠𝑂𝐴𝑃 = 90°

⇒ ∠𝑂𝐵𝑃 = 90°

⇒ ∠𝑂𝐴𝑃 + ∠𝑂𝐵𝑃 = 90° + 90°…….. (𝑖)

⇒ = 180°

In a quadrilateral 𝐴𝑂𝐵𝑃 the sum of all angles = 360°

So ∠𝑂𝐴𝑃 + ∠𝑂𝐵𝑃 + ∠𝐴𝑃𝐵∠ + ∠𝐴𝑂𝐵 = 360°

⇒ 180° + ∠𝐴𝑃𝐵 + ∠𝐴𝑂𝐵 = 360° [𝑓𝑟𝑜𝑚 (𝑖)]

⇒ ∠𝐴𝑃𝐵 + ∠𝐴𝑂𝐵 = 360° − 180°

⇒ ∠𝐴𝑃𝐵 + ∠𝐴𝑂𝐵 = 180°

Hence, the sum of opposite angles are supplementary.

JEE

NEET

Foundation JEE

Foundation NEET

CBSE