NCERT Solutions for Class 10 Maths Chapter 6 Triangles

You can opt for Chapter 6 - Triangles NCERT Solutions for Class 10 Maths PDF for Upcoming Exams and also You can Find the Solutions of All the Maths Chapters below.

NCERT Solutions for Class 10 Maths

• Chapter 1 - Real Numbers

• Chapter 2 - Polynomials

• Chapter 3 - Pair of Linear Equations in Two Variables

• Chapter 4 - Quadratic Equations

• Chapter 5 - Arithmetic Progressions

• Chapter 6 - Triangles

• Chapter 7 - Coordinate Geometry

• Chapter 8 - Introduction to Trigonometry

• Chapter 9 - Some Applications of Trigonometry

• Chapter 10 - Circles

• Chapter 11 - Constructions

• Chapter 12 - Areas Related to Circles

• Chapter 13 - Surface Areas and Volumes

• Chapter 14 - Statistics

• Chapter 15 - Probability

Triangles

The Chapter 6 Maths Triangles of Class 10 NCERT Syllabus is divided into six sections and five exercises where the last exercise section is optional. The first section is the basic introduction to triangles, the figures similar to a triangle and few activities that you can perform to form triangles. The third section consists of the similarity of triangles, few activities, theorems related to a triangle, and followed by an exercise at the end of the section. The fourth section consists of criteria for similarity of triangles, theorems related to the criteria for similarity of triangles, few solved examples, and it ends with an exercise. The fifth section consists of the topic related to the area of similar triangles, theorems related to the area of similar triangles, and ends with an exercise. The sixth section and the final section consists of information related to the Pythagoras Theorem and the theorems and example sums related to the same. After this section comes the optional exercise section where extra sums are provided for you to solve and be through with the concepts. 

Triangles

List of Exercise and Topics Covers in this Chapter:

• Exercise 6.1 : Similar Figures. This exercise has 3 sums

• Exercise 6.2 : Questions related to the Similarity of Triangles. This exercise has 10 sums.

• Exercise 6.3 : Questions related to the criteria for Similarity of Triangles. This exercise has 10 sums.

• Exercise 6.4 : Questions related to Area of similar triangles. This exercise has 14 sums.

• Exercise 6.5 : Questions related to Pythagoras Theorem. This exercise has 17 sums.

• Exercise 6.6 : Extra Questions. This exercise has 10 sums.

In this chapter, we will be learning about triangles and its properties. It contains theorems that will help in solving the problems related to the triangle, and at the end, we have given the summary of the entire chapter.

What do you Need to Know Before you Start Learning this Chapter?

In previous classes, you would have learnt the basics of the triangle. This also includes the properties, calculations of area, finding out one side of the triangle where we have the value of the other two sides of the triangle, and congruency of the triangle in depth. You have to be thorough with these concepts in order to start with Class 10 Maths Chapter 6 Triangles. Since you are done learning these concepts, in this chapter you will be going ahead and learning a little more in-depth about the triangles. You will be understanding the similarities of two triangles of different sizes and you will be able to prove Pythagoras theorem with this knowledge. In addition to that, you will also be learning how to calculate the distance between two objects by indirect measurement. This type of calculation is based on the principle of similarity of figures. 

What are Similar Figures?

You must have learnt that if the radius of the two circles is equal, then it is congruent. Not just that, this was applied to squares if they had the same sides, and equilateral triangles if all the sides of the triangles were equal. These figures were used to check congruence between them. Now, when it comes to the similarity of figures, we look at whether the figures look similar to each other or not. We do not consider facts like the length, height, or the size of the figure. To understand this better, look at the figures given below. 

In these shapes, we are looking for similarity. The circles are the same in shape but different in size. This proves that the circles are similar to each other. The same goes for the square, the triangle and the trapezium. All figures that are congruent to each other can be similar but figures that are similar cannot be congruent to each other. 

Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion). 

The Similarity of Triangles

What do we know about the similarity of the triangles? They need to look alike but do not have to be of the same size. We also do know that the triangle is a polygon. Two triangles are considered to be similar, if:

(i) The corresponding angles of the triangles are equal and

(ii) The corresponding sides of the triangle are in the same proportion or same ratio. 

When two angles of corresponding angles are equal, then these triangles are known as equilateral triangles.

The truth about relating two equilateral triangles was given by a famous Greek mathematician Thales. He said, 

• The ratio of any two corresponding sides in two equiangular triangles is always the same.

• It is believed that he had used a result called the Basic Proportionality Theorem (now known as the Thales Theorem) for the same.

The Important Theorems in this Section are:

Theorem 1: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Theorem 2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

What are the criteria for the triangles to be similar to each other?

Previously, we learnt that two triangles are similar if and only if ( i )  the angles that are corresponding to each other are congruent and ( ii ) the ratio or the proportion of the corresponding sides is the same. 

The important theorems in this section are:

Theorem 3: If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.

This can be referred to as the angle-angle-angle criteria or A-A-A criterion of the two similar angles. This theorem is proved using two triangles and comparing all the angles of the triangles.

If three angles of one triangle are equal to two angles of another triangle, then the two triangles are similar to each other. 

Theorem 4: If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.

This can be referred to as the side - side - side criteria or S-S-S criterion of the two similar angles. This theorem is proved using two triangles and comparing all the angles of the triangles.  If three sides of one triangle are equal to two sides of another triangle, then the two triangles are similar to each other.

Theorem 5: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

This can be referred to as the side - angle - side criteria or S-A-S criterion of the two similar angles. This theorem is proved using two triangles and comparing two of the angles of the triangles and one of the sides of the two triangles. If two sides of one triangle are equal to two sides of another triangle and one side of the triangle is equal to one side of the other triangle, then the two triangles are similar to each other.

How would you calculate the areas of similar triangles?

Now that you have learnt about the criteria, do you think there is a possibility the ratios of the corresponding sides of the triangle are the same? This is true and it can be proved with a theorem. 

Theorem 6: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 

Why is the Pythagoras Theorem used?

Pythagoras theorem is used to prove some activities and solve some problems. You have already learnt the proof of this theorem in 9th grade and this chapter you will learn how to prove the similarity of the triangles using the Pythagorean theorem. The important theorems in this section are: 

Theorem 7: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

This above theorem is used to determine the Pythagoras Theorem. The Pythagoras Theorem is stated as follows: 

Theorem 8: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The Indian mathematician Baudhayan gave out the Pythagoras Theorem. He stated the theorem in the following format: “The diagonal of a rectangle produces by itself the same area as produced by both sides (i.e., length and breadth).” Hence, this theorem is also called Baudhayana Theorem. 

It is also important to know the converse of the theorem. This can be proved in the form of a theorem. The theorem is stated as given below.

Theorem 9: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Now that you have understood the important points in the chapter, you should also know that there is a pattern in the examination to test the knowledge of the students and to check their capabilities to perform better in future. When you understand the NCERT Class 10 Triangles chapter 6, it helps you understand the points you need to focus on - like the weightage of the chapter, the number of questions appearing in the exam - and helps you prepare better for your exam. Practising these NCERT solutions helps you speed up solving the problems and helps you keep up with time and ensure you finish your entire paper within the given duration.

As they say, practice makes you perfect, NCERT ensures that you achieve perfection to complete your papers and increase the speed in solving your problems. You will know what kind of problems might come to your exam and the number of questions from this particular question. This will help you strategize better to score better marks and gauge the frequency of the questions. Most of the time, the questions are usually repeated only the numbers are changed and the questions are a little bit twisted. Thus, solving the previous question papers will help you solve these questions faster. By using the aforementioned points, you will be able to score at least 90% in your exams. 

You can use the NCERT Solutions to:

• Understand your problem-solving skills.

• Understanding the structured and standard questions of the syllabus.

• Improve and understand your Chapter-wise knowledge.

• Increase your confidence level while answering the questions

• Increase your speed while solving sums

• Ensure it serves as your revision notes.

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The Main Strength of the Vedantu’s NCERT Solutions for Class 10  Maths Chapter 6 Triangles, Lies in the Following Points: 

1. Solutions are delivered keeping in mind the age group of the people who would be using them.

2. The solutions are in layman's language and stress on essential facts, terms, principles, theorems, and applications on different ideas.

3. Complicated solutions that are difficult to understand are split into simple parts and well-spaced to save the students from going through all the unnecessary points and strain on their minds.

4. It gives a summary of the entire chapter and concept in the form of solutions.

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Vedantu tried its best to render you real help by providing the NCERT Solutions for Class 10 Maths Chapter 6 maths class 10 Triangles. It aimed to deliver sufficient problems and solutions to practice and build a strong foundation in the chapter. 

What is the Triangles Chapter Important?

Class 10 maths chapter 6 maths Triangles is one of the important topics and can be looked at as a recapitulation of the concept of triangles and congruency of triangles that were discussed in class 9 Math. This chapter has a weightage of 14 marks in class 10 Maths CBSE (board) exams. The average number of questions asked from this chapter is usually seven.

According to the 2020 Paper, the Number of Questions Asked were:

• Section A - 2 questions ( 1 mark )

• Section B - 3 questions ( 2 marks )

• Section C - 2 questions ( 3 marks )

Understanding the Exercises and Sections

Every topic is followed by an exercise which consists of topics that are related to the information given in that section. The reason why there are exercises after every topic is to ensure that you are through with the concept that is being taught. Most of the problems in this chapter are based on theorems that are explained in every section. 

To help you further in understanding these topics and the concepts related to this topic better, a good number of solved examples are explained. In addition to that, step by step explanation of these problems is also shown for every example. This helps in choosing the best approach to solve different types of problems. 

Section 6.1

This section gives an introduction to triangles. It reminds you of what has been taught to you in your previous grades and also explains the prerequisites for this chapter. Some basic definitions, properties and examples are given in this section. Besides that, you will understand that distances can be calculated using indirect means of measurements with the help of similarity of figures. 

Section 6.2 - Exercise 6.1

In this section, you learn mostly about shapes. The difference between similarity and congruency has been explained. The key point explained in this section is how figures can be the same but they do not need to have similar measurements. Two triangles are considered to be similar if: ( i ) The corresponding angles of the triangles are equal and  ( ii ) The corresponding sides of the triangle are in the same proportion or same ratio. When two angles of corresponding angles are equal, then these triangles are known as equilateral triangles.

In exercise 6.1, you have three problems. The first one is fill in the blanks and there are four sub-questions under the first question. The second question asks you to give examples of similar figures and non-similar figures. In the third question you will have to consider two figures and find out whether the two figures are similar to each other or not.

Section 6.3 - Exercise 6.2

Here, you will be learning only about the similarity of the triangle. The important theorems in this section are: ( i ) The corresponding angles of the triangles are equal and ( ii ) The corresponding sides of the triangle are in the same proportion or same ratio. The truth about relating two equilateral triangles was given by a famous Greek mathematician Thales. He said, “The ratio of any two corresponding sides in two equiangular triangles is always the same”. and “It is believed that he had used a result called the Basic Proportionality Theorem (now known as the Thales Theorem) for the same”.

In Exercise 6.2, the questions are mostly based on finding the similarity of triangles.  Some questions have the measurements of the sides of the triangle to compare. And some have ratios. You have to solve some questions using the theorems which have been proved in the above section. To conclude, there are totally ten questions in this exercise.  

Section 6.4 - Exercise 6.3

In the previous section, we stated that two triangles are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion). In this section, you’ll learn about the important criteria. When two angles of corresponding triangles are equal, then these triangles are known as equilateral triangles. ( i ) If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar. This can be referred to as the angle - angle - angle criteria or A-A-A criterion of the two similar angles. This theorem is proved using two triangles and comparing all the angles of the triangles. If three angles of one triangle are equal to two angles of another triangle, then the two triangles are similar to each other.  ( ii ) If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. This can be referred to as the side - side - side criteria or S-S-S criterion of the two similar angles. This theorem is proved using two triangles and comparing all the angles of the triangles.  If three sides of one triangle are equal to two sides of another triangle, then the two triangles are similar to each other. ( iii )Theorem 5:  If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. This can be referred to as the side - angle - side criteria or S-A-S criterion of the two similar angles. This theorem is proved using two triangles and comparing two of the angles of the triangles and one of the sides of the two triangles.  If two sides of one triangle are equal to two sides of another triangle and one side of the triangle is equal to one side of the other triangle, then the two triangles are similar to each other. 

In exercise 6.3, you will have to solve 17 questions and all these questions are based on finding the similarity of the triangle using the criteria SAS, SSS, or AAA. Some of the other questions in this section are - finding the side of the triangle or finding the height of the triangle. 

Section 6.5 - Exercise 6.4

In this section, you are learning only about one single theorem and it is related to the area of a triangle. The theorem states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides

Exercise 6.4

There are around 9 questions in these sections. The last two questions are MCQ based questions. The other questions are to find if two equals are equal, or finding the similarity of the triangle, or finding the area of the triangle or trapezium using the theorem.

Section 6.6 - Exercise 6.5

Pythagoras theorem is used to prove some activities and solve some problems. You have already learnt the proof of this theorem in 9th grade and this chapter you will learn how to prove the similarity of the triangles using the Pythagorean theorem. The important theorem in this section states that “If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other”.

Exercise 6.5

Again, in this section, there are around 17 questions which asked you to justify the correct answer. Or in other words, they are MCQ based questions, finding the distance, finding the base of the wall, proving two sides and one angle of two triangles are equal to each other. 

Summary

1. Two figures having the same size but not necessarily the same shape are called similar Figures.

2. All the congruent figures are similar figures but all similar figures are not congruent.

3. Two figures of the same number of sides are similar, if (i) their corresponding angles are equal to each other and (ii) their corresponding sides are in the same ratio or they are in the same proportion as each other

4. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

6. If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion).

7. If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion).

8. If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion).

9. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion).

10. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

11. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.

12. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem).

13. If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.