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NCERT Class 10 Mathematics Exercise 6.1 Solutions: Triangles

Step-by-step working for every question of Exercise 6.1 in NCERT's Class 10 Mathematics book, chapter “Triangles”, written by Shishya. Each question is in our own short words; open the NCERT chapter for the exact wording and figures.

1. Ex 6.1 Q1

   Fill the blanks: (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 their corresponding angles are … and their corresponding sides are … (equal/proportional).

   1. (i) All circles have the same shape but can have different sizes, so they are similar.
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   2. (ii) All squares have equal angles and proportional sides, but their side lengths can differ, so they are similar.
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   3. (iii) Equilateral triangles always have angles of 60° and proportional sides, making them similar.
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   4. (iv) For polygons to be similar, their corresponding angles must be equal.
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   5. (iv) For polygons to be similar, their corresponding sides must be proportional.
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   Answer: (i) similar; (ii) similar; (iii) equilateral; (iv) equal, proportional

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2. Ex 6.1 Q2

   Give two examples each of (i) similar figures (ii) non-similar figures.

   1. Similar figures have the same shape: corresponding angles equal and corresponding sides in one ratio.
   2. (i) Two circles of radii 2 cm and 5 cm are similar (every circle has the same shape).
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   3. (i) Two equilateral triangles of sides 3 cm and 7 cm are similar (all angles are 60° and the sides are in the ratio 3 : 7).
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   4. (ii) A square of side 4 cm and a rectangle of 4 cm × 6 cm are not similar: the angles match but the sides are not in one ratio.
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   5. (ii) A square and a rhombus of the same side (with angles 60° and 120°) are not similar: the sides match but the angles do not.
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   Answer: (i) two circles; two equilateral triangles. (ii) a square and a non-square rectangle; a square and a non-square rhombus. (Other correct examples are possible.)

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3. Ex 6.1 Q3

   Are the two quadrilaterals drawn in the book (Fig. 6.8, a square and a rhombus of the same side) similar?

   Uses a figure printed in the NCERT book — open the chapter above to see it.

   1. Let the square be PQRS and the rhombus be ABCD, both with side length a.
   2. In a square, all four angles are equal to 90°.
   3. In a rhombus, opposite angles are equal, but adjacent angles are not necessarily 90°.
   4. For two polygons to be similar, their corresponding angles must be equal and their corresponding sides must be in the same ratio.
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   5. The sides of both figures are equal (ratio 1:1), so the side condition is satisfied.
   6. However, the angles of the square (90°) do not match the angles of the rhombus unless the rhombus is also a square.
   7. Since the rhombus in the figure is not a square, its angles are not all 90°.
   8. Therefore, the corresponding angles are not equal, and the quadrilaterals are not similar.
      why?

      In similar triangles the corresponding angles are equal and the corresponding sides are in the same ratio, so one known ratio gives the unknown side. Two equal angles (AA) are enough to show triangles are similar.

   Answer: No, the two quadrilaterals are not similar because their corresponding angles are not equal.

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