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Question
If AB is a chord of a circle, P and Q are the two points on the circle different from A and B, then
Options
∠APB = ∠AQB
∠APB + ∠AQB = 180° or ∠APB = ∠AQB
∠APB + ∠AQB = 90°
∠APB + ∠AQB = 180°
Solution
∠APB + ∠AQB = 180° or ∠APB = ∠AQB
We are given AB is a chord of the circle; P and Q are two points on the circle different from A and B.
We have following figure.
Case 1: Consider P and Q are on the same side of AB
We know that angle in the same segment are equal.
Hence, ∠APB = ∠AQB
Case 2: Now consider P and Q are on the opposite sides of AB
In this case we have the following figure:
Since quadrilateral APBQ is a cyclic quadrilateral.
Therefore,
∠APB + ∠AQB = 180° (Sum of the pair of opposite angles of cyclic quadrilateral is 180°.)
Therefore, ∠APB = ∠AQB or ∠APB + ∠AQB = 180°
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