Chords of equal length cut off equal central angles, and the converse (Theorems 2–3)

What to assume they know

• Turning an observation into a definition: the circle as a locus — circle as a locus; centre, radius, chord; the angle a chord subtends at the centre and how to name it
• Total rotational symmetry, and why every diameter is an axis of reflection — rotational symmetry about the centre
• The SSS and SAS congruence criteria, and what "corresponding parts" means once a congruence is established
• That a triangle with two equal sides has two equal base angles
• Writing a proof as Given / To show / argument

What they should be able to do

• Explain why the triangle formed by any chord and the centre is isosceles, and name its equal sides
• State the equal-chords result and its converse, and say which is which
• Prove that chords of the same length make equal angles at the centre, using SSS
• Prove the converse, using SAS
• Say why the two proofs cannot use the same congruence criterion, given what is supplied in each case
• Identify, in each proof, which pair of angles is the corresponding pair the congruence delivers
• Explain what the rotating-wheel and stretched-arms pictures each contribute, and what they do not establish
• Show that two isosceles triangles built on the radii of one circle are congruent when their bases match

Where it usually goes wrong

• "The wheel picture is the proof." It is the motivation. The chapter states outright that it will now explain why the equality holds, and it is emphatic two pages later, at the top of p. 103, that examples do not establish a general claim. Rotation shows you what to expect; the congruence is what earns it.
• "One theorem, stated two ways." Theorems 2 and 3 are genuinely different claims with different givens, and they need different congruence criteria. A student who thinks a statement automatically implies its converse will fail the concyclicity work in §5.8, where a converse is proved by contradiction because nothing easier is available.
• "Equal chords must be in the same position." They can sit anywhere on the circle; the theorem is precisely that position is irrelevant to the central angle. Fig. 5.8's three wheels are the same chord in three places.
• "The chord and its central angle are proportional." They are not. Doubling the central angle does not double the chord — at radius 10, a 60° angle gives a chord of 10 and a 120° angle gives about 17.3, not 20. The correspondence is one-to-one, which is all the theorems claim.
• "SSS and SAS are interchangeable." They are not, and which one is available is decided by what the given is. That is the cleanest illustration in the chapter of choosing a tool to fit the data.
• "The angle at the centre belongs to the chord alone." It belongs to the chord and this circle. The same 12 cm chord makes a different central angle on a circle of a different radius. Every theorem in this section is stated inside one circle.
• "The stretched-arms story proves the converse." It arranges the claim so you believe it. It quietly assumes the rotation lands the second arm exactly on D, which is the thing being proved.

Questions to check understanding

• Prove that equal chords of a circle make equal angles at the centre
• Prove the converse
• Show that a chord together with the centre always gives an isosceles triangle (the form Exercise Set 5.2 Q1 takes)
• Given a radius and a central angle, find the chord length — including the 60°-gives-the-radius case, which boards ask directly
• Given two chords stated to be equal, mark the equal angles on a figure and justify
• Identify which congruence criterion applies to a stated set of givens, and why the other does not
• One-mark: does doubling a chord's central angle double the chord?

Examples worth working on the board

The chapter prints no answers, so anything marked verified is worked out here on the chapter's own inputs.

• Fig. 5.8 (p. 98, caption names chords and radii; read as the printed page). Three photographs of a spoked bicycle wheel. On each, the same shape is drawn in red over the photograph: two radii running out from the hub to the rim, plus the chord joining their two rim ends — a red triangle. The three wheels differ only in where that triangle sits: upper left on the first, on the right on the second, lower left on the third. That is the figure's entire argument and it is carried by lettering-free artwork, so anyone working from extracted text will not see it at all. The three positions are one thread in three rotated positions.
• Fig. 5.9 (p. 99). Circle with centre C. Points A and B on the upper left, D below and to the left of C with E to its right, so chord DE runs from the lower left across to the right; chords AB and DE drawn, and all four radii CA, CB, CD, CE drawn, so two triangles share the vertex C. Given: AB = DE. To show: the angles ACB and DCE are equal.
• Fig. 5.10 (p. 99). The converse's picture. Circle with centre C; A on the left, B at the bottom, D on the right, E at the top. Two angles are marked in type inside the figure: a = 54° at the centre between the arms to A and B, and b = 54° at the centre between the arms to D and E. Those two printed values are the chapter's only numerical data in this section, and they exist to make "the same angle in a new position" concrete before the proof begins. Give them to the explanation.
• Fig. 5.11 (p. 100). The converse's proof figure. Circle with centre C, E at the top, D on the right, A on the left, B at the bottom; chords AB and ED drawn with the four radii. Given: the angles ACB and DCE are equal. To show: AB = ED.
• The proof inputs, stated as data rather than as the explanation. For Theorem 2: CA = CB = r and CD = CE = r, plus the given AB = DE; three pairs of equal sides, so SSS applies to triangles CAB and CDE, and the wanted angles are the corresponding angles at C. For Theorem 3: CA = CB = r and CD = CE = r, plus the given equality of the two angles at C, which sits between the two known sides in each triangle; so SAS applies and the third sides match.
• Exercise Set 5.2 (p. 100). Q1: show that the triangle a chord makes with the centre is isosceles. Q2: show that two such triangles with equal bases are congruent. Verified: Q1 is immediate — the two sides from the centre are both radii. Q2 is Theorem 2 restated in triangle language, which is worth pointing out: the exercise is not new work, it is the theorem with the circle taken away.
• A numerical hook the explanation can build for section 10 (not in the chapter): radius 10, chord 12. Verified: every chord of length 12 on that circle makes the same angle at the centre, about 73.7°; and every chord making an angle of 60° at the centre has length 10, equal to the radius. The second of these is the form Exercise Set 5.6 Q1 on p. 110 takes, with radius 12 and a central angle of 60° giving chord 12.

Figures to have open

• Fig. 5.8's three wheels, or a redraw. If redrawn, the two radii and the chord must stay one visually connected red shape, and the three copies must differ only by rotation — that is the whole content. This is the chapter's own artwork (p. 98) and the marking is inside the photographs.
• Fig. 5.9 and Fig. 5.11 redrawn as clean schematics with all four radii ticked equal and the two central angles marked. These are the chapter's own figures (pp. 99, 100).
• Fig. 5.10 redrawn with its two printed 54° labels kept. The numbers are the section's only data and dropping them costs the explanation its one concrete anchor.
• A side-by-side proof panel for section 9 showing which parts are given and which are deduced in each direction. Standard schematic; not in the book.
• A non-proportionality figure for the misconception list: on one circle, chords at 60° and 120° at the centre, with their lengths marked. Standard schematic; not in the book.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 5, "I’m Up and Down, and Round and Round", §5.4, printed heading "Chords and the Angles They Subtend", from the middle of p. 98 through p. 100. Theorem 2 with its argument is on p. 99; Theorem 3 with its argument is on p. 100.
• Fig. 5.8 with its caption, p. 98. Figs. 5.9 and 5.10, p. 99. Fig. 5.11, p. 100.
• Exercise Set 5.2, Q1–Q2, p. 100.
• Related exercise elsewhere in the chapter: Exercise Set 5.6 Q1, p. 110, gives a central angle of 60° on a radius of 12 cm and asks for the chord.
• Forward pointer inside the chapter: the chapter's explicit refusal to accept example-based evidence is at the top of p. 103.
• Chapter Summary, p. 117 — the bullet pairing the equal-chords result with its converse.