This page lists every numbered theorem from the circles chapter of the NCERT Class 9 Mathematics textbook — Chapter 5, whose printed title is “I’m Up and Down, and Round and Round” — in book order, with the book’s own statement and the printed page for each.
Every page number below is a printed book page of the current NCERT edition, so you can match each theorem to your own copy. Each row links to a dedicated proof page. The proofs themselves live on those pages — this list is your map, not a substitute for the book.
All Ten Class 9 Maths Theorems in Book Order
The table below lists every numbered theorem of the circles chapter, exactly as the printed book states it.
| Theorem | What the book states | Book page | Page status |
|---|---|---|---|
| Chapter 5: I'm Up and Down, and Round and Round | |||
| Theorem 1 — A Unique Circle Passes Through Three Non-Collinear Points | There is a unique circle passing through three non-collinear points. | 96 | Available |
| Theorem 2 — Equal Chords Subtend Equal Angles at the Centre | Equal chords of a circle subtend equal angles at the centre of the circle. | 99 | Available |
| Theorem 3 — Chords Subtending Equal Angles at the Centre Are Equal | Chords of a circle that subtend equal angles at the centre are equal. | 100 | Available |
| Theorem 4 — Line from the Centre to the Midpoint of a Chord Is Perpendicular | The line joining the centre of a circle and the midpoint of a chord of the circle is perpendicular to the chord. | 101 | Available |
| Theorem 6 — Equal Chords Are Equidistant from the Centre | Chords of a circle having the same length are all at the same distance from the centre of the circle. | 103 | Available |
| Theorem 8 — The Longer Chord Is Nearer the Centre | Let AB and DE be two chords of a circle with centre C. Suppose AB > DE. Then the distance from C to AB is less than the distance from C to DE. | 105 | Available |
| Theorem 9 — Angle at the Centre Is Double the Angle on the Circle | The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc at any point on the circle outside the arc. | 108 | Available |
| Theorem 10 — Equal Angles on the Same Side Mean the Points Are Concyclic | If a line segment AB joining two points A, B subtends equal angles at two other points C, D that lie on the same side of AB, then the four points lie on a circle. | 111 | Available |
| Theorem 11 — Opposite Angles of a Cyclic Quadrilateral Add Up to 180° | The sum of two opposite angles of a cyclic quadrilateral is \(180^\circ\). | 112 | Available |
| Theorem 12 — Converse of the Cyclic Quadrilateral Theorem | If two opposite angles of a quadrilateral add up to \(180^\circ\), then the vertices of the quadrilateral lie on a circle, i.e., they are concyclic. | 113 | Available |
How to Use This Theorem List
Use this list the way you would use the book’s contents page: find your theorem, check its printed page, and open the linked page to see the full proof with all steps. The ten theorems fall into three natural families, and revising them family by family is faster than going theorem by theorem.
The three families of Class 9 circle theorems
- The chord theorems — Theorem 4 (the line from the centre to the midpoint of a chord is perpendicular to the chord), Theorem 6 (equal chords are equidistant from the centre) and Theorem 8 (the longer chord is nearer the centre). These results are all about distances inside the circle, and their proofs build on each other.
- The equal chord and equal angle pair — Theorem 2 (equal chords subtend equal angles at the centre) with its converse, Theorem 3 (chords that subtend equal angles at the centre are equal). Learn these two together because one is the reverse of the other.
- The arc and cyclic quadrilateral group — Theorem 1 (a unique circle through three non-collinear points), Theorem 9 (the angle at the centre is double the angle on the circle), Theorem 10 (equal angles on the same side of a segment mean the four points are concyclic), Theorem 11 (opposite angles of a cyclic quadrilateral add up to 180°) and Theorem 12 (its converse). This is the group that connects angles to circles.
Which theorems are converses of each other?
Three theorems in this chapter are the exact reverse of another result. The pairings below show them side by side so you can see the pattern: the converse swaps the “if” and the “then” of the original statement.
| Original theorem | Its converse | What changes |
|---|---|---|
| Theorem 2 — equal chords subtend equal angles at the centre (p. 99) | Theorem 3 — equal angles at the centre come from equal chords (p. 100) | Start with equal chords; conclude equal angles. Reverse it: start with equal angles; conclude equal chords. |
| Theorem 11 — opposite angles of a cyclic quadrilateral add up to 180° (p. 112) | Theorem 12 — if opposite angles add up to 180°, the quadrilateral is cyclic (p. 113) | Start with a cyclic quadrilateral; conclude the angle sum. Reverse it: start with the angle sum; conclude the quadrilateral is cyclic. |
| Theorem 9 — angle at the centre is double the angle at the circle (p. 108) | Theorem 10 — equal angles on the same side of a segment make the points concyclic (p. 111) | Theorem 10 is the converse-style condition that follows from the double-angle result: the double angle forces the four points to lie on one circle. |
Why Theorem 5 and Theorem 7 are missing
You will notice the list jumps from Theorem 4 to Theorem 6, and from Theorem 6 to Theorem 8. That is not a mistake in this directory. The printed book does not label any statement as Theorem 5 or Theorem 7 in this chapter, so no page can be listed for them.
The ten numbered theorems the book actually states are the ten rows in the table above.
Common mistakes to avoid
Three slips cost more marks in this chapter than any other. The table below names each one and the rule that fixes it.
| Common mistake | Correct rule | How to check your answer |
|---|---|---|
| Using the chord midpoint theorem (Theorem 4) in the wrong direction. | Theorem 4 says: centre + midpoint of a chord → line is perpendicular to the chord. It does not say any perpendicular from the centre lands on the midpoint unless you first prove it. | Ask yourself: do I know the point is the midpoint? If not, the perpendicular-from-centre conclusion has not been established. |
| Applying the double-angle theorem (Theorem 9) at a point on the wrong arc. | The angle at the circumference must be taken at a point on the circle outside the arc, on the side that does not contain the centre for the standard statement. | Mark the arc, then place the point on the other side of the arc from the centre. If the point lies on the arc itself, the theorem does not apply. |
| Citing Theorem 11 where the question needs its converse, Theorem 12. | Theorem 11 takes a cyclic quadrilateral and gives the angle sum. Theorem 12 takes the angle sum and proves the quadrilateral is cyclic. They answer different questions. | Read what the question asks you to prove. “Show the quadrilateral is cyclic” needs Theorem 12; “find the missing angle” needs Theorem 11. |
Sources and Data Verification
This listing is maintained for the 2026-27 academic session using the NCERT textbook information available to us. NCERT remains the authority for confirming the latest edition.
- Which book: the NCERT Class 9 Mathematics textbook, current NCERT edition, official text on ncert.nic.in.
- What this page covers: the numbered theorems of the circles chapter (Chapter 5, printed pages 96 to 113). It does not cover theorems from other chapters or other subjects.
- Maintained for: the current academic session, using the NCERT edition now published.
- Which authority settles what: NCERT settles the textbook, its editions and its page numbers; CBSE settles the curriculum and examinations. Textbook contents and the examinable syllabus are not always identical — check the current official CBSE syllabus.
Common Questions About Class 9 Maths Theorems
Where in the NCERT Class 9 Maths book are these theorems?
All ten are in Chapter 5, “I’m Up and Down, and Round and Round,” the circles chapter, on printed pages 96 to 113. Use the page numbers in the table to find each one in your printed copy.
Why are Theorem 5 and Theorem 7 missing from the table?
Because the printed NCERT book does not label any statement with those numbers. The chapter numbers its results 1, 2, 3, 4, 6, 8, 9, 10, 11 and 12, so this directory lists exactly those ten.
Are the theorem statements on this page the book’s own words?
Yes. Each row carries the statement exactly as the printed book states it, and the page number where it appears, so you can compare the two directly.
Do I need to study the theorems and their converses separately?
Yes. A converse is a different statement with a different proof. Theorem 2 needs Theorem 3, and Theorem 11 needs Theorem 12, as separate revisions — you cannot assume one follows from the other without proof.
Are the NCERT Class 9 theorems the same as the exam syllabus?
Not necessarily. NCERT publishes the textbook; CBSE publishes the examinable Class 9 syllabus. Class 9 mathematics is school-assessed, so check the official CBSE syllabus to confirm what is examinable this session.
Reference: NCERT Class 9 Mathematics textbook, official edition on ncert.nic.in.