In Fig. 10.89, if tangents PA and PB are drawn to a circle such that ∠APS = 30° and chord AC is drawn parallel to the tangent PB, then ∠ABC =

Question

In Fig. 10.89, if tangents PA and PB are drawn to a circle such that ∠ APS = 30 ° and chord AC is drawn parallel to the tangent PB, then ∠ ABC =

Philoid · Accepted Answer

Given:

APB = 30°

Property 1: If two tangents are drawn to a circle from one external point, then their tangent segments (lines joining the external point and the points of tangency on circle) are equal.

Property 2: Sum of all angles of a triangle = 180°

By property 1,

PA = PB (tangent from P)

And,

∠PAB = ∠PBA [∵PA = PB]

By property 2,

∠PAB + ∠PBA + ∠APB = 180°

⇒ ∠PAB + ∠PBA + 30° = 180°

⇒ ∠PAB + ∠PBA = 180° - 30°

⇒ ∠ PAB + ∠ PBA = 150°

⇒ ∠ PBA + ∠ PBA = 150° [∵∠PAB = ∠PBA]

⇒ 2∠PBA = 150°

⇒ ∠PBA = 75°

Now,

∠PBA = ∠CAB = 75° [Alternate angles]

∠PBA = ∠ACB = 75° [Alternate segment theorem]

Again by property 2,

∠CAB + ∠ACB + ∠CBA = 180°

⇒ 75° + 75° + ∠CBA = 180°

⇒ 150° + ∠CBA = 180°

⇒ ∠CBA = 180° - 150°

⇒ ∠CBA = 30°

Hence, ∠CBA = 30°