NCERT Solutions for Class 10 Maths Chapter 6 Triangles Exercise 6.1#
This exercise introduces the concept of similar figures and the criteria for similarity of triangles. Students learn to distinguish between congruent and similar figures, understand that similar figures have the same shape but not necessarily the same size, and apply the AA, SAS, and SSS similarity criteria.
1. Fill in the blanks using the correct word given in brackets: #
(i) All circles are ___. (congruent, similar)
All circles are similar.
Key Concept: Two figures having the same shape (but not necessarily the same size) are called similar figures. All circles are similar because they all have the same shape — a round shape — regardless of their radius.
(ii) All squares are ___. (similar, congruent)
All squares are similar.
All squares have the same shape (all angles are 90° and all sides are equal in ratio), so all squares are similar to each other.
(iii) All ___ triangles are similar. (isosceles, equilateral)
All equilateral triangles are similar.
In equilateral triangles, all angles are 60° each. Since corresponding angles are equal, all equilateral triangles are similar by the AA similarity criterion.
(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are ___ and (b) their corresponding sides are ___. (equal, proportional)
Two polygons of the same number of sides are similar, if:
- (a) their corresponding angles are equal
- (b) their corresponding sides are proportional
Both conditions must be satisfied simultaneously for polygons to be similar.
2. Give two different examples of pair of: #
(i) Similar figures
Example 1: Two circles with radii 2 cm and 4 cm are similar figures.
Example 2: Two squares with sides 3 cm and 6 cm are similar figures.
In both cases, the figures have the same shape but different sizes.
(ii) Non-similar figures
Example 1: A circle and a square are non-similar figures (different shapes).
Example 2: A right-angled triangle and an equilateral triangle are non-similar figures (different angle measures).
3. State whether the following quadrilaterals are similar or not: #
Answer
The two quadrilaterals shown are a square (with all sides equal and all angles 90°) and a rectangle (with all angles 90° but sides of different lengths).
Not similar.
Reason: Although the corresponding angles of a square and a rectangle are equal (both 90°), the corresponding sides are not proportional. In a square, all sides are equal, but in a rectangle, adjacent sides are unequal. Therefore, the condition of proportionality of sides is not satisfied, so the quadrilaterals are not similar.
Key Concept: For two polygons to be similar, BOTH conditions must hold: corresponding angles must be equal AND corresponding sides must be proportional. Satisfying only one condition is not sufficient.