Question

In the given figure, PAT is tangent to the circle with centre O at point A on its circumference and is parallel to chord BC. If CDQ is a line segment, show that:

1. ∠BAP = ∠ADQ
2. ∠AOB = 2∠ADQ
3. ∠ADQ = ∠ADB

Answer 1

i. Since PAT || BC

∴ ∠PAB = ∠ABC (Alternate angles) ...(i)

In cyclic quadrilateral ABCD,

Ext ∠ADQ = ∠ABC  ...(ii)

From (i) and (ii)

∠PAB = ∠ADQ

ii. Arc AB subtends ∠AOB at the centre and ∠ADB at the remaining part of the circle.

∴ ∠AOB = 2∠ADB

`=>` ∠AOB = 2∠PAB  ...(Angles in alternate segments)

`=>` ∠AOB = 2∠ADQ  ...(Proved in (i) part)

iii. ∴ ∠BAP = ∠ADB  ...(Angles in alternate segments)

But ∠BAP = ∠ADQ  ...(Proved in (i) part)

∴ ∠ADQ = ∠ADB