MathematicsChapter 10: Tangents and Angles of a Circle
Alternate Segment Theorem
The angle between a tangent and a chord equals the angle in the alternate (other) segment of the circle.
Learning objectives
- Explain alternate segment theorem using the correct definitions, conditions, and notation.
- Apply tangent-chord = angle in alternate segment correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
The angle between a tangent and a chord equals the angle in the alternate (other) segment of the circle. If the tangent-chord angle is 46 deg, the inscribed angle in the alternate segment is 46 deg. This is not 'half the angle' and not automatically 90.
Formula
angle = 46 deg
Where / symbols
- ALTERNATE SEGMENT THEOREM
- The angle between a tangent and a chord equals the angle in the alternate (other) segment of the circle. If the tangent-chord angle is 46 deg, the inscribed angle in the alternate segment is 46 deg. This is not 'half the angle' and not automatically 90.
- THE EQUALITY
- Tangent at P, chord PA. The angle between tangent and PA equals the angle subtended by PA in the segment on the other side of the chord.
- USE WITH A DIAGRAM
- Mark the tangent, mark the chord, mark the far inscribed angle that 'sees' the same chord from the other segment. That pair is equal. Combine with isosceles radii if needed.
Relationship
- tangent-chord = angle in alternate segment
Overview
Look across the chord:
The Equality
Tangent at P, chord PA. The angle between tangent and PA equals the angle subtended by PA in the segment on the other side of the chord.
Examples 46 deg -> 46 deg · Not 23 deg · Not 90 unless it happens to be
Use With A Diagram
Mark the tangent, mark the chord, mark the far inscribed angle that 'sees' the same chord from the other segment. That pair is equal. Combine with isosceles radii if needed.
Examples Name the chord · Name the far angle · Then write equal
WHAT — ALTERNATE SEGMENT THEOREM
The angle between a tangent and a chord equals the angle in the alternate (other) segment of the circle. If the tangent-chord angle is 46 deg, the inscribed angle in the alternate segment is 46 deg. This is not 'half the angle' and not automatically 90. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Alternate Segment Theorem matters because tangent and circle-angle theorems explain the geometry at a point of contact. They support proofs, numerical angle questions, and practical circle constructions.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through THE EQUALITY, then USE WITH A DIAGRAM. Worked: tangent-chord angle at P is 46 deg. Angle in the alternate segment? alternate segment theorem those two angles are equal inscribed angle = 46 deg Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- tangent-chord angle at P is 46 deg. Angle in the alternate segment?
- alternate segment theorem
- those two angles are equal
- inscribed angle = 46 deg
In simple words
Alternate segment is not the alternate angle of parallel-line Chapter 9 of Class 9. It is a circle theorem with a similar English word.
Key points
- tangent-chord = angle in alternate segment
- equal angles
- same chord, other segment
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Alternate Segment Theorem
Sticky Notes · Quick Revision
Core Idea
- The angle between a tangent and a chord equals the angle in the alternate (other) segment of the circle. If the tangent-chord angle is 46 deg, the inscribed angle in the alternate segment is 46 deg. This is not 'half the angle' and not automatically 90.
THE EQUALITY
- Tangent at P, chord PA. The angle between tangent and PA equals the angle subtended by PA in the segment on the other side of the chord.
USE WITH A DIAGRAM
- Mark the tangent, mark the chord, mark the far inscribed angle that 'sees' the same chord from the other segment. That pair is equal. Combine with isosceles radii if needed.
Worked Method
- Worked: tangent-chord angle at P is 46 deg. Angle in the alternate segment?
- alternate segment theorem
- those two angles are equal
- inscribed angle = 46 deg
Common Misconception
- Alternate segment is not the alternate angle of parallel-line Chapter 9 of Class 9. It is a circle theorem with a similar English word.
FBISE Exam Check
- Alternate segment is not the alternate angle of parallel-line Chapter 9 of Class 9. It is a circle theorem with a similar English word.
60-second revision
- The angle between a tangent and a chord equals the angle in the alternate (other) segment of the circle.
- THE EQUALITY: Tangent at P, chord PA.
- USE WITH A DIAGRAM: Mark the tangent, mark the chord, mark the far inscribed angle that 'sees' the same chord from the other segment.
- Worked-method reminder: inscribed angle = 46 deg
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.