CBSE Class 10 Mathematics Chapter 6: Triangles NCERT Solutions

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NCERT Solutions PDF Class 10 PDF

This chapter focuses on the concept of similarity in geometric figures, particularly triangles. Students will learn to differentiate between similar and congruent figures and understand the conditions required for polygons, especially triangles, to be considered similar. The solutions cover exercises that involve filling in blanks with appropriate terms like 'similar' or 'congruent', identifying pairs of similar and non-similar figures, and determining if given quadrilaterals are similar based on their angles and side lengths. These NCERT Solutions provide clear explanations and step-by-step guidance, aiding students in grasping fundamental geometric principles and preparing effectively for their examinations.

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Quick info

BoardCBSE
ClassClass 10
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 21

Chapter summary

Chapter 6, Triangles, for Class 10 Mathematics NCERT Solutions introduces the concepts of similarity and congruence. It clarifies the conditions under which figures, especially triangles and polygons, are deemed similar. The exercises focus on distinguishing between similar and congruent shapes, providing examples of both, and applying the criteria for similarity to quadrilaterals. This chapter builds a foundational understanding of geometric similarity, crucial for further studies in geometry.

Learning outcomes

  • Understand the difference between similar and congruent figures.
  • Identify conditions for similarity in polygons and triangles.
  • Provide examples of similar and non-similar geometric figures.
  • Determine similarity of quadrilaterals based on angles and sides.
  • Apply the concept of similarity to solve basic geometric problems.

Topics covered

Paper topics

  • Similar Figures
  • Congruent Figures
  • Polygons
  • Triangles
  • Corresponding Angles
  • Corresponding Sides
  • Proportional Sides
  • Equal Angles
  • Equilateral Triangles
  • Squares
  • Circles
  • Quadrilaterals

Important topics

  • Conditions for Similarity
  • Distinguishing Similar vs. Congruent Figures
  • Similarity in Triangles
  • Similarity in Polygons
  • Application to Quadrilaterals

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Questions and Solutions

Exercise 6.1

Question 1

Fill in the blanks using the correct word given in the 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:

The relationship between geometric figures is determined by their shape and size. Similarity implies the same shape, while congruence implies the same shape and size.

(i) All circles are **similar**. This is because all circles have the same round shape, regardless of their radius. They can be scaled to match each other.

(ii) All squares are **similar**. All squares have four equal sides and four right angles (90 degrees). Thus, they share the same shape and can be scaled to match each other.

(iii) All **equilateral** triangles are similar. An equilateral triangle has three equal sides and three equal angles, each measuring 60 degrees. This consistent shape ensures all equilateral triangles are similar.

(iv) Two polygons of the same number of sides are similar if:

(a) their corresponding angles are **equal**.

(b) their corresponding sides are **proportional** (i.e., in the same ratio).

Question 2

Give two different examples of a pair of: (i) Similar figures (ii) Non-similar figures
Solution:

(i) Examples of similar figures:

  1. Two equilateral triangles: Consider an equilateral triangle with side length 1 cm and another equilateral triangle with side length 2 cm. Both have angles of 60 degrees, and their sides are in the ratio 1:2.
  2. Two squares: Consider a square with side length 1 cm and another square with side length 2 cm. Both have four right angles, and their sides are in the ratio 1:2.

(ii) Examples of non-similar figures:

  1. A square and a rectangle (that is not a square): A square has all angles equal to 90 degrees and all sides equal. A rectangle has all angles equal to 90 degrees, but its adjacent sides can have different lengths. If the sides are not in proportion, they are not similar. For example, a square with side 2 cm and a rectangle with sides 2 cm and 4 cm.
  2. A circle and a square: These figures have fundamentally different shapes. A circle is defined by a radius and has no straight sides or angles, while a square has four straight sides and four right angles.

Question 3

State whether the following quadrilaterals are similar or not: Quadrilateral PQRS with sides PQ = 1.5 cm, QR = 1.5 cm, RS = 1.5 cm, SP = 1.5 cm and angles P = 90°, Q = 90°, R = 90°, S = 90°. Quadrilateral ABCD with sides AB = 3 cm, BC = 3 cm, CD = 3 cm, DA = 3 cm and angles A = 90°, B = 90°, C = 90°, D = 90°.
Solution:

To determine if two quadrilaterals are similar, we need to check two conditions:

  1. Are their corresponding angles equal?
  2. Are their corresponding sides proportional?

Let's examine the given quadrilaterals:

Quadrilateral PQRS: All sides are 1.5 cm, and all angles are 90 degrees. This is a square.

Quadrilateral ABCD: All sides are 3 cm, and all angles are 90 degrees. This is also a square.

Now let's check the conditions for similarity:

1. Corresponding Angles:

  • Angle P = 90° and Angle A = 90°
  • Angle Q = 90° and Angle B = 90°
  • Angle R = 90° and Angle C = 90°
  • Angle S = 90° and Angle D = 90°

All corresponding angles are equal (90°).

2. Corresponding Sides:

Let's find the ratio of corresponding sides:

  • PQ / AB = 1.5 cm / 3 cm = 1/2
  • QR / BC = 1.5 cm / 3 cm = 1/2
  • RS / CD = 1.5 cm / 3 cm = 1/2
  • SP / DA = 1.5 cm / 3 cm = 1/2

All corresponding sides are proportional, with the ratio being 1:2.

Since both conditions (equal corresponding angles and proportional corresponding sides) are met, the two quadrilaterals (both squares) are **similar**.

Common mistakes

  • Confusing similarity with congruence.
  • Incorrectly applying conditions for similarity (e.g., assuming angles are equal when only sides are proportional, or vice versa).
  • Difficulty in providing distinct examples of similar and non-similar figures.

Revision tips

  • Clearly define and differentiate between 'similar' and 'congruent' figures.
  • Memorize the conditions for similarity: equal corresponding angles and proportional corresponding sides.
  • Practice identifying similar and non-similar figures with varied examples.
  • Pay close attention to the details of angles and side lengths when assessing quadrilateral similarity.

Practice MCQs

Q1. What is the relationship between all equilateral triangles?

Q2. Two figures are considered similar if:

Q3. Which of the following pairs of figures are always similar?

Q4. For two polygons to be similar, their corresponding angles must be:

Q5. If two polygons have proportional corresponding sides, are they necessarily similar?

Frequently asked questions

What is the main difference between similar and congruent figures in Class 10 Maths?

Congruent figures have the same shape and the same size, meaning they can be superimposed exactly onto each other. Similar figures have the same shape but can have different sizes; their corresponding angles are equal, and their corresponding sides are in the same ratio.

What are the conditions for two triangles to be similar?

Two triangles are similar if (i) their corresponding angles are equal, and (ii) their corresponding sides are in the same ratio (proportional).

Are all circles similar?

Yes, all circles are similar because they have the same shape (round) and their corresponding angles are equal (though circles don't have angles in the same way polygons do, the concept of scaling applies uniformly). Their radii can be in any proportion.

How can I use these NCERT Solutions for Chapter 6 Triangles?

These solutions provide step-by-step explanations for each question in Exercise 6.1. Use them to understand the concepts of similarity, check your answers, and learn the methods for identifying similar figures and applying the conditions of similarity.

What does it mean for sides to be 'proportional' in similar figures?

Sides are proportional when the ratio of the lengths of corresponding sides in two similar figures is constant. For example, if triangle ABC is similar to triangle PQR, then AB/PQ = BC/QR = AC/PR.

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