The two conditions polygons must both meet

What to assume they know

• What similarity asks for that congruence does not — that similarity holds shape fixed and lets size go
• Naming corresponding vertices of two polygons and keeping the pairing straight
• Writing a ratio of two lengths and testing whether several ratios are equal
• The angle properties of a square, a rectangle and a rhombus from earlier classes
• That the interior angles of any quadrilateral total 360°
• Reading a scale on a map or a plan

What they should be able to do

• State the two requirements a pair of polygons must both meet to be called similar, and say which one is about angles and which about lengths
• Compute the scale factor for a pair of polygons from any one matched pair of sides, and check it against the remaining pairs
• Show, from the printed measurements of Fig. 6.5, that a stated pair meets both requirements
• Use the square-and-rectangle pair to show that matching angles alone are not enough
• Use the square-and-rhombus pair to show that a common side ratio alone is not enough
• Explain what the light-and-shadow activity constructs, and why rays from one point produce a figure of the same shape
• Write down the vertex correspondence for a similar pair, and explain what goes wrong if the correspondence is scrambled
• Decide whether a given quadrilateral pair is similar, given the four sides and the angle marks

Where it usually goes wrong

• "Equal angles is the real test; sides follow." The rectangle kills this. Four right angles against four right angles, and the pair is still not similar.
• "Equal side ratios is the real test; angles follow." The rhombus kills this. Every ratio equal to 2, and the pair is still not similar.
• "So the two clauses are just being thorough." They are not. Each is independently necessary, and the section spends two figures proving it.
• "A square and a rhombus are both diamonds, so they are the same shape." Equal sides do not fix a quadrilateral. A rhombus can be squashed flat while keeping every side length — which is precisely why the angle clause is needed.
• "The scale factor may drift a little from side to side." One number, all pairs. 6/7 and 1 are not near enough to each other to count.
• "Vertices can be paired however you like." The similarity claim includes the pairing. Pair the wrong corners and a similar pair can be made to look like a failure, and occasionally the reverse.
• "The shadow is bigger, so the shadow is a different shape." Light travels in straight lines from one point, which is exactly the construction that scales without distorting.

Questions to check understanding

• Fill in the two clauses of the definition from memory, in the right order
• Given two quadrilaterals with all sides and angles marked, decide similarity and justify by naming which clause fails
• Given a scale factor and one figure's sides, produce the other figure's sides
• Construct a counter-example to "equal angles implies similar", and one to "proportional sides implies similar"
• Name the scale factor for a stated pair, and then state the factor the other way round
• One-mark recall: the other name for the scale factor used on maps and plans
• Explain why a square set beside a rhombus with matching side ratios still fails the test

Examples worth working on the board

Values marked verified are worked out here on measurements printed inside pp. 73–98. The chapter prints no answers.

• The enlargement that motivates the definition (p. 75). A 35 mm negative printed at 45 mm, or at 55 mm. Every segment is scaled by one number; every inclination between a pair of segments is untouched. Those two observations become the two clauses.
• Activity 1 and Fig. 6.4 (p. 76, checked). A single lamp burns overhead at a ceiling point named O; a cardboard quadrilateral ABCD is held flat and level somewhere between that lamp and a table below; the shadow it throws is traced on the table top. Verified by opening the page: the shadow is lettered with primes — A′B′C′D′ — and the primes are lost entirely by text extraction, which printed pages both figures as ABCD. The drawing also carries two children holding the card. Each shadow vertex lies on the ray from O through the matching card vertex: A′ on ray OA, B′ on OB, C′ on OC, D′ on OD. The activity then asks the student to measure and find the four angle equalities and the four equal ratios AB/A′B′, BC/B′C′, CD/C′D′, DA/D′A′.
• Fig. 6.5, a pair that passes (p. 77, checked). Quadrilateral ABCD: AB = 1.5 cm, BC = 2.5 cm, CD = 2.4 cm, DA = 2.1 cm, with ∠A = 105°, ∠B = 100°, ∠C = 70°, ∠D = 85°. Quadrilateral PQRS: PQ = 3.0 cm, QR = 5.0 cm, RS = 4.8 cm, SP = 4.2 cm, with ∠P = 105°, ∠Q = 100°, ∠R = 70°, ∠S = 85°. Verified: the four side ratios are 3.0/1.5, 5.0/2.5, 4.8/2.4 and 4.2/2.1 — every one of them exactly 2, so the scale factor is 2 and the reverse scale factor is ½. Verified: the four angles in either figure total 105 + 100 + 70 + 85 = 360°, as a quadrilateral's must.
• Fig. 6.6, angles only (p. 77, checked). Square ABCD with every side 3 cm; rectangle PQRS with SR = PQ = 3.5 cm and SP = RQ = 3 cm. Right-angle marks are drawn at all four corners of each. Verified: both figures have four right angles, so the angle clause holds perfectly, but the side ratios come out 3/3.5 = 6/7 ≈ 0.857 on one pair and 3/3 = 1 on the other. Two different numbers, so there is no scale factor, so the pair is not similar.
• Fig. 6.7, sides only (p. 78, checked). Square ABCD with every side 2.1 cm; rhombus PQRS with every side 4.2 cm. Neither is given a right-angle mark here — the drawing carries side lengths and nothing else. Verified: all four side ratios are 4.2/2.1 = 2, a perfect scale factor — and the pair is still not similar, because the square's angles are right angles and the rhombus is drawn slanted, so no angle pairing can be made to agree.
• Exercise 6.1 (p. 78). Question 1 runs to four items but five blanks, because the last item has two, with the choices supplied in brackets: (i) circles, choosing between congruent and similar; (ii) squares, between similar and congruent; (iii) which triangles are always similar, between isosceles and equilateral; (iv) the two clauses themselves, one blank each, filling equal and proportional in the right order. Verified: the five fillings are similar, similar, equilateral, and equal then proportional. Question 2 asks for two examples each of a similar pair and a non-similar pair.
• Exercise 6.1 question 3 and Fig. 6.8 (p. 78, checked). Two quadrilaterals to be judged: PQRS with all four sides 1.5 cm and drawn as a slanted rhombus carrying no right-angle marks, and ABCD with all four sides 3 cm and a right-angle mark drawn at each of its four corners. Verified: every side ratio is 3/1.5 = 2, and the angles cannot be matched, so the pair is not similar. This is the Fig. 6.7 counter-example again at a different size, which is worth saying out loud — the exercise is checking that the student learned the counter-example rather than the pictures.
• Transitivity (p. 77). The remark that similarity chains: whenever one polygon matches a second, and that second matches a third, the match carries all the way from the first to the third. Verified as arithmetic: scale factors multiply, so a factor of 2 followed by a factor of 3 is a factor of 6.

Figures to have open

• The bulb-and-shadow apparatus of Fig. 6.4 (p. 76): point source O above, a quadrilateral card held level below it, the shadow outline on the table, and the four rays drawn through from O. This is the chapter's own figure and the argument of sections 5 and 6 depends on the rays being visible; redraw it as a clean schematic rather than reproducing the printed art.
• Fig. 6.5's pair (p. 77) redrawn with every one of the eight sides and eight angles carrying its printed value.
• Fig. 6.6's square-and-rectangle pair and Fig. 6.7's square-and-rhombus pair (pp. 77–78), each redrawn with its measurements. Fig. 6.6's pair carries a right-angle mark at every corner of both quadrilaterals and the redraw should keep them; Fig. 6.7 carries none at all, on either shape, and the redraw should not invent any. These two are the load-bearing figures of the topic.
• Fig. 6.8's pair (p. 78) for the closing exercise section.
• A scale bar on a simple map or floor plan, for section 4. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 6 "Triangles", §6.2 Similar Figures, pp. 75–78, including Activity 1 (p. 76) and the Remark on chaining (p. 77). Figures 6.4, 6.5, 6.6, 6.7.
• Exercise 6.1, p. 78, questions 1 to 3, with Fig. 6.8.
• The same two clauses are restated for triangles at the head of §6.3 (p. 79) and again at the head of §6.4 (p. 85); point 3 of the chapter's summary (p. 97) is the compressed form.