Question

Two circles intersect each other at two points a and

b. From a, tangents ap and aq to the two circle are drawn which intersect the circles at the points p and q respectively. Prove that ab is the bisector of angle pbq

Answer 1

hope this will help you

Answer 2

Proved below.

Step-by-step explanation:

Given:

As shown in the figure, two circles intersect each other at two points a and b.

Also from point a, tangents ap and aq to the two circle are drawn which intersect the circles at the points p and q respectively.

Now,

∠ abq = ∠ qax           [ Alternate interior angles]

∠ abp = ∠ yap               [ Alternate interior angles]

As, ∠ yap = ∠ qax         [ vertically opposite angles ]

⇒ ∠ abp = ∠ abq          [1]

Also,

∠ abq + ∠ abp = 180   [ sum of linear pair angles is 180 ]  

So,

∠ abq + ∠ abp = 180

∠ abq + ∠ abq = 180   [ from 1 ]

2 ∠ abq = 180

$\angle abq=\frac{180}{2}$

∠ abq = 90

Similarly,

∠ abq + ∠ abp = 180

∠ abp + ∠ abp = 180     [from 1]

2∠ abp = 180

$\angle abp = \frac{180}{2}$

∠ abp = 90

Therefore, ∠ abq = ∠ abp = 90

Hence ab is the bisector of angle pbq.

Hence proved.