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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.
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Alternate Segment Theorem
What is it?
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.
In simple words
- 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
Important terms
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.
What you should learn
- 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.
Examples
- 46 deg -> 46 deg
- Not 23 deg
- Not 90 unless it happens to be
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.
Recommended for Class 10
Full school-level explanation
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.
Look across the chord:
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.
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.
Remember
- equal angles
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.
Remember
- same chord, other segment
Examples
- Name the chord
- Name the far angle
- Then write equal
Remember
- tangent-chord = angle in alternate segment
- 46 deg
- circle, not parallel lines
- equal angles
- same chord, other segment
- tangent-chord = angle in alternate segment
- tangent-chord = angle in alternate segment
- mathematical reasoning
How the ideas connect
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
Deeper understanding
Go beyond memorising Alternate Segment Theorem. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
THE EQUALITY
USE WITH A DIAGRAM
Worked Method
Exam focus
Quick revision
- tangent-chord = angle in alternate segment
- 46 deg
- circle, not parallel lines
- equal angles
- ALTERNATE SEGMENT THEOREM: The angle between a tangent and a chord equals the angle in the alternate (other) segment of the cir
- THE EQUALITY: Tangent at P, chord PA. The angle between tangent and PA equals the angle subtended by PA in the seg
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.
For long answers on Alternate Segment Theorem: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- tangent-chord = angle in alternate segment
- 46 deg
- circle, not parallel lines
- equal angles
Important formulas
- tangent-chord = angle in alternate segment
- 46 deg
- circle, not parallel lines
Common exam traps
- 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.
- Not 23 deg
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
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.
- Mistake
Not 23 deg
- Mistake
Not 90 unless it happens to be
- Mistake
Do not treat THE EQUALITY and USE WITH A DIAGRAM as the same step; each has a different purpose.
- Mistake
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.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
tangent-chord = angle in alternate segment
46 deg
circle, not parallel lines
equal angles
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 ang
tangent-chord = angle in alternate segment
- 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 ang
- Formula
- tangent-chord = angle in alternate segment
- Example
- 46 deg -> 46 deg
- Critical warning
- Alternate segment is not the alternate angle of parallel-line Chapter 9 of Class 9. It is a circle theorem with a simila
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Exam Practice
Define
Q1. Define Alternate Segment Theorem.
Explain
Q2. Explain: THE EQUALITY.
Example
Q3. Give one clear example of Alternate Segment Theorem and say why it fits.
Mistake
Q4. State one common mistake students make about Alternate Segment Theorem.
Exam note
Q5. Write a short exam note on Alternate Segment Theorem: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.