Exercise 6.1 — Triangles

1. 12 marksNCERT Cl-10 Maths, Ex 6.1, Q1

   Fill in each blank with the correct word from the bracket. (i) All circles are ______ (congruent, similar). (ii) All squares are ______ (similar, congruent). (iii) All ______ triangles are similar (isosceles, equilateral). (iv) Two polygons with the same number of sides are similar if (a) their corresponding angles are ______ and (b) their corresponding sides are ______ (equal, proportional).

   Hint. Ask each time whether the shape is forced to be the same. Size is allowed to vary for similarity; it is not allowed to vary for congruence.

   (i) similar. Every circle has the same shape — a circle is completely described by its radius, and changing the radius only scales it. Two circles of different radii are therefore similar but not congruent, so "congruent" would be too strong.

   (ii) similar. Same reasoning. Every square has four right angles and four equal sides, so any two squares differ only by a scale factor. They are congruent only in the special case where the sides happen to be equal.

   (iii) equilateral. In an equilateral triangle all three angles are forced to be 60°, so any two equilateral triangles are equiangular and their sides are automatically in a constant ratio. Isosceles triangles are not: one with apex 20° and one with apex 100° are both isosceles and clearly not the same shape.

   (iv) (a) equal, (b) proportional. This is the definition of similarity for polygons — corresponding angles equal and corresponding sides in the same ratio.

   ✦ Answer: (i) similar (ii) similar (iii) equilateral (iv) (a) equal, (b) proportional

   Where students slip. Answering "congruent" for (i) and (ii). All circles are the same *shape*, not the same *size* — congruence would require every circle to have the same radius, which is obviously false.

   Another way. For (iii), test the claim rather than recalling it: draw two isosceles triangles with different apex angles. They fail on sight, which leaves equilateral as the only option.

2. 22 marksNCERT Cl-10 Maths, Ex 6.1, Q2

   Give two different examples of a pair of (i) similar figures, (ii) non-similar figures.

   Hint. For similar, pick shapes whose form is fixed by a single number. For non-similar, deliberately break exactly one of the two conditions.

   (i) Two pairs of similar figures.

   • Any two circles — say one of radius 2 cm and one of radius 5 cm. Shape is fixed; only the scale changes.
   • Any two equilateral triangles — say of side 3 cm and side 7 cm. All angles are 60° in both, and the sides are in the constant ratio 3 : 7.

   A third good answer: a photograph and its enlargement, which is the textbook's own opening example.

   (ii) Two pairs of non-similar figures.

   • A square and a rectangle (say 4 cm × 4 cm and 4 cm × 7 cm). All eight angles are right angles, so the angle condition holds — but 4/4 ≠ 4/7, so the side condition fails.
   • A circle and a square. There is no correspondence of sides or angles at all, so they cannot be similar.

   A third: a scalene triangle and an equilateral triangle, since their angles cannot be matched up.

   ✦ Answer: similar — two circles; two equilateral triangles. Non-similar — a square and a rectangle; a circle and a square.

   Where students slip. Offering "two triangles" or "two quadrilaterals" as examples of similar figures. Being triangles is not enough — the shape has to be forced, which is why *equilateral* triangles work and general ones do not.

   Another way. Choosing a square and a rhombus for part (ii) makes a sharper point: there the *sides* are proportional and it is the *angles* that fail, showing that either condition can be the one that breaks.

3. 32 marksNCERT Cl-10 Maths, Ex 6.1, Q3 (Fig. 6.8)

   The figure shows two quadrilaterals: a square ABCD with every side 3 cm and all four angles marked as right angles, and a rhombus PQRS with every side 1.5 cm whose angles are not right angles. State whether the two quadrilaterals are similar.

   Hint. Check the two conditions separately. One of them passes here — that is the trap.

   Step 1 — Test the sides. Every side of ABCD is 3 cm and every side of PQRS is 1.5 cm, so each pair of corresponding sides is in the ratio 3 : 1.5 = 2 : 1 The ratio is the same for all four pairs, so the side condition holds.

   Step 2 — Test the angles. ABCD is a square, so each of its angles is 90°. PQRS is a rhombus that is not a square, so its angles are not 90° — a rhombus has two acute and two obtuse angles. So the corresponding angles are not equal, and the angle condition fails.

   Step 3 — Both conditions are required. Similarity of polygons needs equal angles and proportional sides. Passing one test is not enough, which is exactly what the textbook demonstrates with its square-and-rhombus pair just before this exercise.

   ✦ Answer: No — the quadrilaterals are not similar. Their sides are proportional, but their corresponding angles are unequal.

   Where students slip. Concluding "similar" the moment the four side ratios come out equal. For triangles one condition does imply the other, but for quadrilaterals and other polygons it does not — that distinction is the whole point of this question.

   Another way. A one-line argument: a square has a right angle and a non-square rhombus does not, so no correspondence of vertices can make the angles match. The side lengths never need to be computed.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 10 Mathematics textbook, Reprint 2026-27 (chapter 6 now runs to three exercises — 6.1, 6.2 and 6.3; the proof of Pythagoras' theorem and the theorem on areas of similar triangles, with old exercises 6.4, 6.5 and 6.6, are no longer part of this chapter). Questions are referenced from the NCERT textbook for identification.