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Chord

If A B is a c...

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

(a) ∠APB = ∠AQB

(b) ∠APB + ∠AQB = 180° or ∠APB = ∠AQB

(c) ∠APB + ∠AQB = 90°

(d) ∠APB + ∠AQB = 180°

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Solution

(b) ∠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°

Hence, the correct answer is option (b).

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If AB is a chord of a circle, P and Q are the two points on the circle different from A and B, then (a) ∠APB = ∠AQB (b) ∠APB + ∠AQB = 180° or ∠APB = ∠AQB (c) ∠APB + ∠AQB = 90° (d) ∠APB + ∠AQB = 180°

If AB is a chord of a circle, P and Q are the two points on the circle different from A and B, then

(a) ∠APB = ∠AQB

(b) ∠APB + ∠AQB = 180° or ∠APB = ∠AQB

(c) ∠APB + ∠AQB = 90°

(d) ∠APB + ∠AQB = 180°