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Mathematics · Class 10 · Chapter 28: Angle in a Segment of a Circle

8 min

Key points

An angle in a segment is an angle formed by two chords meeting at a point on the circle
Angles in the same segment of a circle are equal
The angle in a semicircle is always 90° (right angle)
Angle in major segment is acute (less than 90°)
Angle in minor segment is obtuse (greater than 90°)
The angle at the center is twice the angle at the circumference standing on the same arc
Opposite angles of a cyclic quadrilateral are supplementary (sum = 180°)
If a quadrilateral is inscribed in a circle, it is called a cyclic quadrilateral

Definitions

Angle in a Segment: An angle in a segment of a circle is the angle formed by two chords drawn from any point on the arc of the segment to the endpoints of the chord forming the segment.
Cyclic Quadrilateral: A quadrilateral whose all four vertices lie on the circumference of a circle is called a cyclic quadrilateral.
Segment of a Circle: The region between a chord and the arc subtended by it is called a segment of the circle.
Arc: A part of the circumference of a circle is called an arc.

Worked example

In a circle with center O, points A, B, C, D lie on the circumference such that AB is a chord. If angle ACB = 35°, find angle ADB. Justify your answer using angle in a segment theorem. (5 marks)

Angle ADB = 35°. By the angle in a segment theorem, angles subtended by the same chord in the same segment of a circle are equal.

Common mistakes

Students confuse angle in segment with angle at center
Forgetting to state 'angles in same segment are equal' before using the property
Not drawing proper diagrams with labeled points and arcs
Mixing up major and minor segments when identifying angle types
Not mentioning that opposite angles in cyclic quadrilateral are supplementary

Quick recap

Angle in a segment theorem states that angles subtended by the same chord in the same segment of a circle are equal. This applies only when both angles are on the same side of the chord. Special case: angle in semicircle is always 90°. Always state the theorem exactly as in STBB textbook for full marks.

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