Question

In the picture below, PQ, RS, TU are tangents to the circle at A, B, C.

How many pairs of equal angles are there in it?

Answer 1

In a circle, each angle between a chord and the tangent at one of its ends is equal to the angle in the segment on the other side of the chord.

So, the pair of equal angles are:

∠BAC = ∠UCB …(1)

∠BAC = ∠SBC …(2)

∠ABC = ∠TCA …(3)

∠ABC = ∠PAC …(4)

∠ACB = ∠QAB …(5)

∠ACB = ∠RBA …(6)

From (1) and (2):

∠UCB = ∠SBC

From (3) and (4):

∠TCA = ∠PAC

From (5) and (6):

∠QAB = ∠RBA

So, there are nine pairs of equal angles in the given figure.