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
Answers
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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
∠ abq = 90
Similarly,
∠ abq + ∠ abp = 180
∠ abp + ∠ abp = 180 [from 1]
2∠ abp = 180
∠ abp = 90
Therefore, ∠ abq = ∠ abp = 90
Hence ab is the bisector of angle pbq.
Hence proved.