What is the inscribed angle theorem?

The inscribed angle theorem states that the angle subtended by an arc at the centre of a circle is double the angle subtended by the same arc at any point on the remaining part of the circle. Example: if arc PQ subtends 60° at a point on the circle, it subtends 120° at the centre. Special case: when the arc is a semicircle (180° at centre), the inscribed angle = 90° — this is the 'angle in a semicircle' theorem.

What is a cyclic quadrilateral?

A cyclic quadrilateral is a quadrilateral whose all four vertices lie on a circle. The key property is that opposite angles are supplementary: ∠A + ∠C = 180° and ∠B + ∠D = 180°. Examples include rectangles (all angles 90°, sum = 180°) and isosceles trapeziums. The converse is also true: if opposite angles of a quadrilateral are supplementary, it is cyclic.

How are angles in the same segment equal?

If two angles are inscribed in a circle and both subtend the same chord from the same side (same segment), they are equal. This is because both equal half the central angle subtending that chord. Example: if AB is a chord, all angles ∠ACB where C is on the major arc will be equal (they all equal half the central angle ∠AOB).

Angles Subtended by Arcs and Chords

Circles • Class 9 Mathematics • NCERT • CBSE

Key angle theorems: (1) Angle at centre = 2 × angle at circumference (same arc). (2) Angles in the same segment are equal. (3) Angle in a semicircle = 90°. (4) Opposite angles of a cyclic quadrilateral are supplementary (sum = 180°).

Key Formulas

Angle at centre = 2 × angle at circumference (same arc)
Angles in same segment are equal
Angle in semicircle = 90°
Cyclic quadrilateral: opposite angles sum = 180°

Frequently Asked Questions

What is the inscribed angle theorem?: The inscribed angle theorem states that the angle subtended by an arc at the centre of a circle is double the angle subtended by the same arc at any point on the remaining part of the circle. Example: if arc PQ subtends 60° at a point on the circle, it subtends 120° at the centre. Special case: when the arc is a semicircle (180° at centre), the inscribed angle = 90° — this is the 'angle in a semicircle' theorem.
What is a cyclic quadrilateral?: A cyclic quadrilateral is a quadrilateral whose all four vertices lie on a circle. The key property is that opposite angles are supplementary: ∠A + ∠C = 180° and ∠B + ∠D = 180°. Examples include rectangles (all angles 90°, sum = 180°) and isosceles trapeziums. The converse is also true: if opposite angles of a quadrilateral are supplementary, it is cyclic.
How are angles in the same segment equal?: If two angles are inscribed in a circle and both subtend the same chord from the same side (same segment), they are equal. This is because both equal half the central angle subtending that chord. Example: if AB is a chord, all angles ∠ACB where C is on the major arc will be equal (they all equal half the central angle ∠AOB).

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