NCERT Solutions Class 10 Maths Chapter 6 Triangles : Exercise 6.1

NCERT Solutions Class 10 Maths Chapter 6 Exercise 6.1explores the solutions to the first exercise of Chapter 6. The exercise is all about the basic concepts of triangles and their similarity in geometry. This exercise lays the foundation for the similarity of figures in geometry, which is essential for solving complex problems related to geometry and real life. Here, we have broken down the complete exercise as per the CBSE syllabus and examination. So, let’s dive deeper into the exercise to grasp the key concepts of geometry.

1.0Download NCERT Solutions Class 10 Maths Chapter 6 Triangles Exercise 6.1 : Free PDF 

2.0Introduction to Similarity of Polygons: 

Before delving into the topic of triangles, let’s take a quick look at the similarity of a broader category of shapes in geometry, i.e., polygons. The similarity of Polygons is used to describe two polygons with the same shape but not necessarily the same sizes. Two polygons are similar if their corresponding angles are equal and their corresponding sides are proportional. For instance, consider a real-life example, a picture of an object printed in different sizes, all of these photographs will be similar because they have the same shape, however different size. 

What is the Similarity of Triangles?

The similarity of triangles refers to pairs of triangles that have the same angles but may differ in size. The concept of similarity is central to indirect measurements, such as estimating the height of mountains or estimating the distance of objects in space. 

Access our NCERT Solutions Class 10 Maths chapter 6 exercise 6.1 PDF, designed by our expert faculty and crafted according to the latest CBSE syllabus and examination pattern.

3.0NCERT Class 10 Maths Chapter 6 Exercise 6.1 Overview: Key Concepts

Conditions for Similarity of Polygons: 

To verify whether two polygons are similar, they must satisfy the following conditions:

• Corresponding Angles are Equal: The angles between the corresponding sides of the two polygons should be equal. For example, in the figure of two triangles, as shown above, 

∠A = ∠X 

∠B = ∠Y 

∠C = ∠Z  

• Corresponding Sides are Proportional: The two polygons' corresponding sides must be in the same ratio. This ratio is referred to as the scale factor. To understand it better, in the figure of triangles shown above, 

$\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}$

Scale Factor in Similar Triangles: 

The scale factor is a crucial concept of similar triangles, which relates to the corresponding sides of similar triangles. The scale factor is also referred to as the representative fraction. It is expressed as the ratio of the length of any two corresponding sides of two similar triangles, and this ratio remains constant for every side of two similar triangles. The scale factor can be expressed as: 

$\text{Scale Factor}=\frac{\text{Length of side one triangle}}{\text{length of corresponding side in other triangle}}$

The scale factor also helps determine the size of other triangles or polygons. For example, if the scale factor is more than 1, the second triangle is an enlargement of the first. If the scale factor is less than 1, the second triangle is a reduction of the first.

Similar and Congruent Figures: 

• Congruent Figures: Two figures are congruent if they are equal in both shape and size. All congruent figures are similar, but similar figures are not always congruent.
• Similar Figures: Two figures are similar if:
• • Their corresponding angles are equal.
  • Their corresponding sides are to the same ratio or proportion. 

Non-Similar Figures: 

A pair of shapes that are similar in shape are considered similar figures. However, there are some examples in which this notion doesn’t work. These examples are: 

• A square and a rectangle are not similar because while their angles are equal, their corresponding sides are not proportional.
• A square and a rhombus are not congruent because although their sides will be proportional, their angles are not equal.

Remark: If given some polygons, say triangles A, B, and C, in these triangles, let triangle A be similar to triangle B and Triangle B be similar to triangle C, then triangle A is similar to triangle C. This property of similar polygons works when there are more than two polygons and is known as transitivity of similarity. 

Explore our NCERT Solutions Class 10 Maths chapter 6 exercise 6.1 PDF Online to get hands-on experience with problems related to the exercise with a detailed explanation.

Download NCERT Solutions Class 10 Maths Chapter 6 Exercise 6.1 for complete solutions and a detailed explanation of each question of this exercise. 

Also Read CBSE Notes Class 10 Maths Chapter 6 Triangles

4.0NCERT Class 10 Maths Chapter 6 Triangles Exercise 6.1 : Detailed Solutions

1. Fill in the blanks using the correct word given in brackets :

(i) All circles are ________ (congruent, similar)

(ii) All squares are ________ . (similar, congruent)

(iii) All ________ triangles are similar.(isosceles, equilateral)

(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)

Solution:

(i) All circles are similar.

(ii) All squares are similar.

(iii) All equilateral triangles are similar.

(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are equal and (b) their corresponding sides are proportional.

2. Give two different examples of pair of

(i) Similar figures.

(ii) Non-similar figures.

Solution:

(i) 1. Pair of equilateral triangles are similar figures.

2. Pair of squares are similar figures.

(ii) 1. One equilateral triangle and one isosceles triangle are non-similar.

2. Square and rectangle are non-similar.

3. State whether the following quadrilaterals are similar or not :

Solution:

The two quadrilaterals in the figure are not similar because their corresponding angles are not equal.

5.0Benefits of NCERT Solutions Class 10 Maths Chapter 6 Triangles Exercise 6.1

• Helps students understand fundamental properties of triangles, including similarity and congruence.
• Offers detailed solutions that make complex problems easier to grasp.
• Strengthens problem-solving skills and builds confidence for board exams.
• Enables students to apply key theorems like Pythagoras’ theorem and similarity criteria.
• Helps students solve problems efficiently by following structured methods.
• Clarifies common mistakes and misconceptions to improve accuracy.