What similarity asks for that congruence does not

What to assume they know

• Congruence of triangles from Class IX, and the fact that congruent figures can be laid one on the other and coincide
• That a circle is fixed by its radius, a square by its side, an equilateral triangle by its side
• Reading a ratio of two lengths as a single number, and simplifying it
• Measuring an angle with a protractor, and knowing that an angle is unchanged when a figure is scaled
• A fixed step between neighbours is the whole definition for the habit of testing a definition against cases rather than accepting it on sight

What they should be able to do

• State the difference between two figures being congruent and being similar, in terms of which of shape and size is being held fixed
• Explain why every congruent pair is a similar pair, and produce a similar pair that is not congruent
• Decide, for circles, squares and equilateral triangles, which pairs from each family are congruent and which are only similar
• Explain why a circle and a square can never be similar, and why a triangle and a square can never be similar
• Identify a pair of figures whose similarity cannot be settled by looking, and say what would have to be measured instead
• Describe what an enlargement from a photographic negative does to lengths and what it leaves alone
• Give an example of two figures that share a size but not a shape, and say why that pair fails the test
• Explain in outline how similarity makes an unreachable height measurable

Where it usually goes wrong

• "Similar means roughly alike." It does not admit any slack at all. The shape must agree exactly; only the size is free. A figure that is nearly the right shape is not similar, it is simply not similar.
• "Congruent and similar are two alternatives to choose between." They are nested. Congruence is the special case in which the scale happens to be 1, so a congruent pair is always also a similar pair.
• "If it looks similar, it is." Fig. 6.2 exists to break this. The page's position is not that the pair fails — it prints no length and no angle, so nobody, reader or author, is in a position to say. What it says is that looking cannot settle the question, and that is why the chapter stops and builds a definition before answering anything.
• "All triangles are similar, since they are all triangles." Only the equilateral ones come with a guarantee. Being in the same family is not enough unless the family is fixed by a single number.
• "Bigger means not similar." Size is exactly the thing similarity does not look at.
• "Enlarging a photograph opens out the angles." It does not. That angles are untouched by scaling is half of what makes the definition work.
• "Similar figures must be the same way up." Nothing in the idea mentions orientation. Fig. 6.1's nested pairs happen to be aligned; that is drawing convenience, not a requirement.

Questions to check understanding

• Fill-in-the-blank on which families are always similar and which are always congruent — Exercise 6.1 question 1 items (i) to (iii) is exactly this
• Give two examples of a similar pair and two of a non-similar pair, with a reason for each — Exercise 6.1 question 2
• True or false with justification: every congruent pair is similar; every similar pair is congruent
• Explain why two figures that look alike cannot be declared similar on that basis
• Short answer on what an enlargement does to lengths and to angles
• A one-mark recall on the name for measuring a height without reaching it

Examples worth working on the board

Values marked verified are worked out here on data printed inside pp. 73–98. This chapter prints no answers of its own.

• The chapter opener (p. 73, checked). A QR code tagged 1062CH06, the chapter title with a boxed numeral 6, and two uncaptioned cartoon panels below the opening paragraphs: a climber roped to a snow peak claiming the mountain's height is easy to measure, and a figure on a rooftop with a ladder claiming they will reach the moon. Neither panel carries a Fig. number. The explanation can use these two claims as the question the chapter answers.
• Fig. 6.1 (p. 74, checked). Three panels, (i) circles, (ii) squares, (iii) equilateral triangles. Verified by opening the page: each panel shows one large figure with a smaller one of the same family drawn nested inside it, plus two further figures standing separately — four members per family. No measurement of any kind is printed on the artwork; the text alone carries the reasoning. In panel (ii) the two free-standing squares are drawn close enough in size that congruence within the family looks like a near miss. Note that the running text makes its point about a family holding both congruent and non-congruent members while discussing the circles of panel (i); of the squares and the equilateral triangles it says only that similarity follows the same way.
• Fig. 6.2 (p. 74, checked). Two quadrilaterals side by side, lettered ABCD (A upper left, D upper right, B lower left, C lower right) and PQRS (P upper left, S upper right, Q lower left, R lower right). Both read as trapezium-like four-sided figures. Verified by opening the page: no side length and no angle is printed on either figure. That absence is the whole point of the section — the reader is given exactly nothing to compute with, and the chapter says so.
• Fig. 6.3 (p. 75, checked). Three black-and-white photographs of the Taj Mahal, each in its own white frame, printed in decreasing sizes left to right. Same monument, same framing, three sizes.
• The photographer's enlargement (p. 75). A negative of 35 mm is enlarged to 45 mm, or to 55 mm. Every line segment scales by the same factor and every angle between a pair of segments is left alone. Verified: the enlargement factor is 45/35 = 9/7 ≈ 1.286 in the first case and 55/35 = 11/7 ≈ 1.571 in the second; running it the other way gives 35/45 = 7/9 ≈ 0.778 and 35/55 = 7/11 ≈ 0.636. Handy for showing it: a segment 21 mm long on the small print becomes 27 mm at 45 mm and 33 mm at 55 mm, and a 40° inclination stays 40° in all three.
• The size-without-shape counter-case (p. 75). Two photographs of one person printed at the same size, one taken at age 10 and one at age 40. Equal size, unequal shape — so the pair fails, and it fails on the half of the test that congruence would have passed.
• What similarity is for (pp. 73–74). Heights of mountains, Mount Everest named, and distances of far objects, the moon named, are obtained by indirect measurement resting on similarity. The chapter points forward to Example 7 and to question 15 of Exercise 6.3 inside this chapter, and to Chapters 8 and 9 of the book — a deliberate cross-reference outside pp. 73–98.

Figures to have open

• The three panels of Fig. 6.1 (p. 74) redrawn as a schematic: four members per family, one nested inside another, two standing free. Must not be traced from the printed art.
• The bare quadrilateral pair of Fig. 6.2 (p. 74), redrawn with no measurements printed.
• Three photographs of one subject at three sizes. A generic monument or building photograph will serve; the textbook's own Taj Mahal images are not required and should not be reproduced.
• A photographic-enlargement schematic: a small frame and a large frame with one corresponding segment marked in each and the shared inclination arc drawn on both. Standard schematic.
• Optional, for section 12: a sight-line diagram to a distant peak. Standard schematic; the chapter's own opener art is cartoon and need not be copied.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 6 "Triangles", §6.1 Introduction, pp. 73–74, and §6.2 Similar Figures, pp. 74–75. Figures 6.1, 6.2 and 6.3.
• Forward pointers made by the chapter itself: Example 7 (p. 92) and question 15 of Exercise 6.3 (p. 97) inside this chapter; Chapters 8 and 9 of the book, which lie outside pp. 73–98 and are cited here as the chapter's own cross-reference.