Q1 of 62 Page 193

In the figure TP is a tangent to a circle. A and B are two points on the circle. If ∠BTP = 72° and ∠ATB = 43° find ∠ABT.

The alternate segment theorem (also known as the tangent-chord theorem) states that in any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment.


Therefore by above theorem,


∠BAT = ∠BTP = 72°


Sum of all angles of a triangle = 180°


In ΔABT,


∠ABT + ∠BTA + ∠TAB = 180°


⇒ ∠ABT + 43° + 72° = 180°


⇒ ∠ABT + 115° = 180°


⇒ ∠ABT = 180° – 115°


⇒ ∠ABT = 65°


Hence, ∠ABT = 65°


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