Question

The number of equal angles an arc subtends in the opposite segment is _______.

• A. two
• B. four
• C. none
• D. infinite

Answer 1

The correct option is D infinite

The angle subtended by an arc in a segment are equal. It can be proved as shown below.

Let AB be the arc, O be the centre of the circle and let $\angle$ACB and $\angle$ADB be the angles subtended by arc AB in the major segment formed by it. Join OA and OB.

Now $\angle$AOB is the angle subtended by arc at the center of the circle which is equal to twice the angle subtended by the arc at any point on the circle.

So, $\angle$AOB = 2$\angle$ACB = 2 $\angle$ADB

Therefore, $\angle$ACB = $\angle$ADB

Here C and D are any general points on the circle. Therefore the number of equal angles subtended by an arc in opposite segment are infinite .