# NCERT Solutions for Class 10 Maths Chapter 6

## NCERT Solutions for Class 10 Maths Chapter 6 - Triangles

NCERT Solutions for Class 10 Maths Chapter 6 Triangles are offered by Vedantu helps in exam preparation. Vedantuâ€™s Solution for NCERT books for Class 10 Maths offers students with detailed study material. This step by step solutions to** CBSE NCERT Class 10 Maths **Chapter 6 provides you with a clear understanding of the subject. You can also opt for Triangles Class 10 NCERT Solutions PDF for your upcoming CBSE Board Exams 2019-20. Download NCERT Solution PDF today to have easy access to all subject solutions for free.

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## NCERT Solutions for Class 10 Maths Chapter 6 - Triangles

NCERT Class 10 Maths Chapter 6 on triangles is a crucial chapter in the Standard 10 curriculum. Here, in this chapter, NCERT introduces you to the different concepts of a triangle, its angles, necessary calculations, etc.Â

Essentially, it is vital to understand this chapter in detail with Vedantuâ€™s NCERT Solutions for Class 10 Maths Chapter 6 to be able to cope with the curriculum of Standards 11 and 12.Â

Our solutions offer concise and comprehensive answers to the exercises of NCERT Maths Class 10 Chapter 6, along with those ofÂ all the other chapters.

This Class 10 Maths NCERT Solutions Chapter 6 helps students complete all the given exercises with standard answers accompanied by relevant shortcut techniques for faster solving. Prepared by expert teachers, this NCERT Solution for Class 10 Maths Chapter 6 along with others effectively aids you in improving your Standard 10th board exam scores.Â

Our accurate NCERT Class 10 Maths Chapter 6 and other solutions are available for easy download sans charges, allowing you to avail them at any time. We, at Vedantu, offer free PDF download for NCERT solutions in all subjects, crucially aiding in your board examination preparation.

**NCERT Solution of Maths Class 10 Chapter 6 â€“ All Exercise Questions.**

The NCERT Maths Class 10 Chapter 6 is divided into six sections, covering all basic concepts of triangles. Beginning with an introduction to triangles and other geometric figures, it goes on into different types of triangles and their properties.Â

Furthermore, included in detail in our NCERT Solution of Class 10 Maths Chapter 6, various calculations and proofs related to the properties of triangles are also covered.Â

The vital questions and topics covered in our free PDF downloads offering NCERT Solutions for Class 10 Maths Ch 6 are discussed below.

I. NCERT Class 10 Maths Chapter 6: Exercise 6.1.

Â Â Â Â a)Â Question 1:

The first question in this chapter checks your understanding of triangles. Different types of triangles and their properties are discussed in our NCERT Solutions for Class 10 Maths Chapter 6 Triangles. You have to fill in the blanks to fit the correct description discussed in these questions.

Â Â Â b)Â Question 2:

This first section of NCERT Class 10 Maths Chapter 6 teaches you a few other geometric shapes and figures along with triangles. Here in this question, you have to provide two examples, each of similar and non-similar figures. These are accompanied by a relevant explanation of their similarity and otherwise in our solutions.

Â Â Â Â c)Â Question 3:

In the final question of this section, you have to study 2 different figures and recognize their properties. You also have to state whether they are similar to each other or not. Our solutions provide the relevant explanations along with the answer.

**NCERT Solution for Class 10 Maths Chapter 6: Exercise 6.2.**

Â Â Â Â Â a) Question 1:

The second section from NCERT Class 10 Maths Chapter 6 explains the different characteristics of a triangle and relevant theorems. Consequently, the first question in this section requires students to find the length of the base with the other measurement provided in the question. You can take the help of our solutions on this exercise to understand the working of the theorems.

Â Â Â Â b) Question 2:

Different measurements of 3 similar triangles are provided in this question. You have to use these measurements to find if a specific lattice is parallel to another line. Our solutions offer a detailed explanation of the proofs of all these problems.

Â Â Â Â c) Question 3:

This question from NCERT Class 10 Maths Chapter 6 tests your understanding of triangles as well as parallel lines. The basic proportionality theorem is the working function in this question.Â

Â Â Â d) Question 4:

This question is similar to the previous one, where students must prove two ratios to be equal. However, here the given figure isÂ more complicated with on the parallel lines meeting a vertex while the other meets a lattice. Students can easily refer to our solutions to understand the method of proving this question.

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Â Â Â e) Question 5:

While the previous proofs involved basic proportionality theorem, this proof requires the converse of it too. While proving this question, you have to use the converse of the basic proportionality theorem in the last step to prove that a specific line in a triangle is parallel to another.Â

Our NCERT Solutions for Class 10 Maths Ch 6 offers a detailed explanation of the use and the logic of this theorem.

Â Â Â Â f) Question 6:

This question is similar to the previous one though here students must work with two triangles. The smaller of the two is placed within the other, making their three lattices parallel lines. Students must prove them to be parallel lines by reasoning.

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Â Â Â g) Question 7:

In this question, you have to prove that a line drawn through mid-points of one side of a triangle bisects the other side. Here, the said line is parallel to the other side of this triangle. Our NCERT Solution Class 10 Maths Chapter 6 explains how this proof becomes crucial to solve other questions.

Â Â h)Â Question 8:

This proof requires the functioning of the converse of the basic proportionality theorem. Here students are required to prove that a line joining midpoints of two sides of a triangle are always parallel to the third side. As mentioned in the NCERT textbook, this question has been previously explained as in our solutions from Class 9.

Â Â i) Question 9:

Quadrilaterals can be divided into sections to represent different triangles. Consequently, in this question, you have to study the figure of a trapezium and subsequently prove two proportions of lattices and intersections to be equal.

Â Â Â j) Question 10:

The last question from Exercise 6.2 of NCERT Class 10 Maths Chapter 6 also consists of a quadrilateral. Interestingly, here, students must prove that this figure is a trapezium. It requires detailed knowledge of quadrilaterals and students can refer to our solutions for aid.

**III.Â ****Maths NCERT Class 10 NCERT Solutions Class 10 Chapter 6: Exercise 6.3.**

Â Â a) Question 1:

Six pairs of triangles are provided in this question from NCERT Class 10 Maths Chapter 6, along with their figures. You are required to prove that these pairs of triangles are similar to each other. Additionally, you must also mention the criterion used to prove their similarity.

Â b) Question 2:

The second question here involves similar triangles in which the values of certain angles are specified. The given figure is of a pair of parallel lines which has two lines intersecting each other, forming an inverted triangle over another. Students can avail of our Maths NCERT Class 10 NCERT Solutions Class 10 Chapter 6 to understand the solving of this question.

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Â Â c) Question 3:

While the previous question had a diagram to it, explaining the different structures and relevant angles; in this question, the figure is just described. You have to use similarity criteria to prove that two proportions of lattices are the same.

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Â Â d) Question 4:

The similarity of specific ratios and proportions of lattices is mentioned in this question, along with the relevant diagram. The proof requires you to show that two triangles formed in this figure are similar to each other. In case of difficulty in demonstrating this proof, students can refer to our NCERT Solutions for Class 10 Maths Chapter 6.

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Â e) Question 5:

Students must demonstrate in this question that two triangles given in the figure are similar to each other. Our solutions to NCERT Class 10 Maths Chapter 6 explain this proof in relevant details along with a diagram.

Â Â f) Question 6:

This problem involves understanding the congruency of triangles. It is given that two triangles are congruent to each other. Consequently, students must find out if two other triangles formed are similar to each other.

Â g) Question 7:

A figure is given in this question, along with the mention of other facts. The figure includes numerous triangles that are formed by the intersection of their altitudes. You are required to show a few specific triangles to be similar to others. Our solutions on NCERT Class 10 Maths Chapter 6 discuss all the proofs.

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Â Â h) Question 8:

A parallelogram is intersected by a line which meets the extension of one arm of this figure. Of the numerous triangles formed, you must prove two of them, to be similar to each other. This question involves the theorem on alternate interior angles.

Â Â i) Question 9:

Students are introduced to the concept of a right-angled triangle. Consequently, in this question, a figure involving right-angled triangles is presented. Students must prove the similarity between the two triangles. Additionally, this question from NCERT Class 10 Maths Chapter 6 also requires students to prove that two proportions are the same.

Â j) Question 10:

The figure of two similar triangles and their properties are discussed in this question. Students must use this given data to prove two different sets of triangles to be similar. Additionally, explained in detail in our NCERT Solutions for Class 10 Maths Chapter 6 PDF, students must also prove two proportions to be equal.

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Â k) Question 11:

An interesting question involving an isosceles triangle, your understanding of perpendiculars is tested here. The figure has two perpendicular lines, one of which intersects the base of a triangle. You must prove one of these triangles formed to be similar to another.Â

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Â l) Question 12:

Two figures are provided in this question. These figures have sides that are proportional to one another. Explained in detail in our NCERT Class 10 Maths Chapter 6 solutions and study material, students must prove the similarity between two triangles.

Â m) Question 13:

This question includes an understanding of theorems and proportions along with the working of proofs. Students must prove that the square of one side is the same as the multiplication of two other sides.

Â Â n) Question 14:

NCERT Class 10 Maths Chapter 6 covers median and relevant theorems of triangles. In this question, students must prove the similarity of two triangles. As demonstrated in our solutions, students must know all the similarity theorems to determine this question.

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Â Â o) Question 15:

This penultimate question of this exercise describes the formation of the shadow of a pole and a tower. With the length of these shadows and the height of the pole being given, students must find the height of the tower.

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Â Â p) Question 16:

The question provides figures of two triangles that are similar to each other. You are required to prove that the proportion of two sides of these triangles is equal to each other. In case stuck while trying to prove this question, you should take the help of our solutions on NCERT Class 10 Maths Chapter 6.

**IV. NCERT Solutions for Class 10 Maths Chapter 6: Exercise 6.4.**

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Â Â a) Question 1:

While the previous section primarily had questions that required you to prove specific characteristics or similarities, in this section you must solve and find the values of relevant lattices, ratios, etc. along with different proofs. In this first question, you are required to find the value of a specific lattice of a triangle. The areas of these triangles are given along with the length of one of their lattices.

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Â Â Â b) Question 2:

In this second question of NCERT Class 10 Maths Chapter 6, students must find the ratio of the areas of two triangles formed by intersections in a trapezium. Additionally, it is also mentioned in the question that one side of the trapezium is double of its other side. Students can refer to our solutions for a better understanding of the solution and relevant theorems.

Â Â c) Question 3:

Another question involving trapeziums, here two triangles are formed by a chord which intersects the quadrilateral. Students must demonstrate with step by step solutions that the area of the two triangles formed is in the same proportion as their medians.

Â Â d) Question 4:

You must prove in this question that if the areas of two similar triangles are the same, they must be congruent with each other. In case you find it difficult to solve this question, avail our solutions to NCERT Class 10 Maths Chapter 6.

Â Â e) Question 5:

A triangle is formed by connecting the mid-points of the vertices of a triangle. While the new triangle formed is similar to the bigger one, you must find the ratio of their area. This question requires knowledge of corresponding angles along with similarity theorems.

Â Â f) Question 6:

This question is detailed proof which requires a thorough understanding of this chapter. Students must prove that the ratio of areas of two similar triangles is equal to the ratio of squares of their medians. In case stuck, students should refer to our solution sets since clarity on these basic concepts is crucial for higher studies.

Â Â g) Question 7:

The question describes two triangles, the former of which is equilateral and formed on one side of a square. The second triangle is formed on the diagonals of this same square. You are required to prove that the area of the triangle discussed above is half of that of the latter. Our Class 10 Maths NCERT Solutions Chapter 6 discusses the proof with relevant theorems.

Â Â h) Question 8:

In this question, students are informed of two equilateral triangles. Additionally, one vertex of one of the triangles is mid-point of the lattice of the other triangle. In this question from NCERT Class 10 Maths Chapter 6, students must find the ratio between the areas of these two triangles.Â

Â Â i) Question 9:

The ratio of the sides of two similar triangles is provided in this question. You are required to find the ratio of their area. Four proportions are mentioned in the question, and students must solve the question to choose the correct option.

**V. NCERT Solution for Class 10 Maths Chapter 6: Exercise 6.5.**

Â Â a) Question 1:

The first question of this exercise in NCERT Class 10 Maths Chapter 6 checks how well you understand triangles and their types. Four different sets of lengths are given, and you must mention which of them are right-angled triangles. In case it is such a triangle, you must also mention the length of the hypotenuse of this triangle.

Â Â b) Question 2:

A right-angled triangle is described in this question. A point â€˜pâ€™ connects its hypotenuse to the right angle. Students must prove that the square of this line is equal to the multiplication of the two segments on the hypotenuse. In case a student is stuck with this proof, it is best to understand the detailed explanation from our solution.

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Â Â c) Question 3:

Though similar to the previous question, this one has more equations for students to prove. A right-angled triangle and its properties are discussed here. Our NCERT Solution of Class 10 Maths Chapter 6 offers a detailed description of the proof on all the equations in the question.

Â Â d) Question 4:

The question from Section 6.5 of NCERT Class 10 Maths Chapter 6 presents an isosceles right-angled triangle. You must prove if the square of this hypotenuse is equal to the double of base square. This problem involves knowledge of different types of triangles along with Pythagoras theorem.

Â Â e)Â Question 5:

An isosceles triangle is presented in this question. Additionally, the question also mentions that two of its sides are equal. It is also mentioned that the square of one side is equal to the double of another sideâ€™s square. Students must prove that this triangle is right-angled.

Â Â f) Question 6:

An equilateral triangle is described here with sides of a given value in variables. Consequently, students must find the altitude of all the sides in this question from NCERT Class 10 Maths Chapter 6. This question tests a studentâ€™s understanding of different types of angles as well as their understanding of the Pythagoras theorem.

Â Â Â g) Question 7:

Rhombus is a type of quadrilateral that has its particular characteristics. Your knowledge of quadrilaterals is checked here along with that of triangles. You must prove that the sum of the squares of its sides is equal to the sum of squares of its diagonals. In case you have any trouble with the numerous theorems being involved, you can opt for Triangles Class 10 NCERT Solutions PDF download for clarity.

Â Â h) Question 8:

Here a figure of a triangle enclosing a square is presented in the question. Numerous characteristics of the different figures in this diagram are mentioned. Students must use this information to prove the two equations mentioned in this question.

Â Â i) Question 9:

This problem discusses a ladder that goes up to reach a window at a considerable height from the ground. In this interesting question from NCERT Class 10 Maths Chapter 6, students must find the distance between the foot of this ladder and the wall. Pythagoras theorem is required to solve this problem.

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Â Â Â j) Question 10:

This question is similar to the previous one though it involves a more complex situation. The situation involves a stake that is tied to a rope. The other end of this rope is tied to a pole. You are required to find how deep the base of this pole must be driven so that the rope is taut.

Â Â Â k) Question 11:

It is an interesting question where two airplanes are discussed. Both leave the airport with specific speeds yet fly in different directions. Students must find out the distance between these two airplanes after a specific period.

Â Â l) Question 12:

Two poles of different heights are fixed at a certain distance from each other in this question from NCERT Class 10 Maths ChapterÂ 6. Students must find the distance between the tips of these two poles. This problem requires an understanding of the Pythagoras theorem along with the characteristics of quadrilaterals and triangles.

Â Â m) Question 13:

A right-angled triangle is described in this question along with specific points marked on its vertices. Accordingly, students must prove an equation of the different vertices and its segments. Referring to our NCERT Solution of Maths Class 10 Chapter 6, you easily understand how to prove the given equation.

Â Â n) Question 14:

In aÂ given triangle, a perpendicular intersects the opposite lattice in a specific proportion. Consequently, students must solve the given equation and prove that it is valid with the information provided with this question.Â

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Â Â o) Question 15:

One of the sides of an equilateral triangle is intercepted by a point, dividing the lattice into a specific proportion. Provided with an equation informing you of the lengths of the intersected arm, you must prove another equation given in this question from NCERT Class 10 Maths Chapter 6.

Â Â p) Question 16:

It is another question based on the concepts of equilateral triangles. Students must prove that three times the square of one side is equal to 4 times the square of its altitude. In case any student has trouble in solving this proof, our solutions on NCERT Class 10 Maths Chapter 6 offer a detailed explanation of the theorem used along with the solving.

Â Â q) Question 17:

This last question of this section provides the length of 3 lattices of a triangle. You must find the value of a specific angle with this given information and tick the correct option accordingly. As explained in our solutions, you must check if this is a right-angled triangle and accordingly find the value of this specific angle.

**VI. NCERT Class 10 Maths Chapter 6: Exercise 6.6.**

Â Â Â a)Â Question 1:

More complex theorems on bisection and resulting proofs are the scope of this section. Consequently, the first question in this last section of NCERT Class 10 Maths Chapter 6 involves bisection, basic proportionality theorem, angle bisectors, etc. You must refer to our solution sets to get a better understanding of the solution to this question.

Â Â Â b) Question 2:

In this question, students must apply their knowledge of right angles and right-angled triangles. A quadrilateral is formed within a triangle in the given figure. From the given figure, students must prove that the square of specific arms of this quadrilateral is equal to the multiplication of two other lattices.

Â Â Â Â

Â Â Â c) Question 3:

The solution to this question requires knowledge of obtuse-angled triangles and their characteristics. Students must prove aÂ complex equation that involves different lattices of the above-discussed triangle. It is advisable to take the aid of our Chapter 6 Maths Class 10 NCERT Solutions to understand the various theorems involved in this proof properly.

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Â Â Â d) Question 4:

In this proof, students have to deal with acute-angled triangles. Similar to the previous question in this section from NCERT Class 10 Maths Chapter 6, students must prove the given equation of different lattices. Its solution involves the Pythagoras theorem along with the understanding of the characteristics of triangles.

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Â Â Â e) Question 5:

Along with the given figure, different proportions and facts of the given figure are mentioned in the question. Here, you have to prove three different equations which are mentioned in the question, using the provided information. It primarily involves Pythagoras theorem.

Â Â Â f) Question 6:

This question is complex, and it is advisable to refer to our solutions on NCERT Class 10 Maths Chapter 6 for a detailed understanding of the same. You are required to prove that the sum of the diagonals of this parallelogram is equal to the sum of the squares of their sides.

Â Â Â g) Question 7:

Here, students require knowledge of chords and their functioning to answer this question accurately. In the given figure of a circle, two chords intersect each other at point P to meet the circle at its circumference. Students must prove that the triangles formed are similar.Â

Additionally, students must also prove that the multiplication of similar lattices of the triangles comes as equal.

Â Â h) Question 8:

Two chords of a circle are extended outside to meet at point P. Two triangles are formed here, where the smaller triangle is a part of the larger one. You must prove that these two triangles formed are similar to each other. This question also requires an understanding of cyclic quadrilaterals.Â

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Â Â i) Question 9:

A figure is presented in this question where you must prove that a particular line is the bisector of an angle from which it is originating. Along with the basic proportionality theorem, students must understand corresponding angles as well as alternate interior angles. You can refer to our NCERT Class 10 Maths Chapter 6 solutions to understand the solution better in case you find it challenging to solve.

Â Â j) Question 10:

It is the last question of this chapter. Here, a scenario involving fishing has been discussed. The distance between the water and this fishing rod, the distance of the fly, etc. are mentioned. Accordingly, with all the given information, students must find the distance of the fly after a specific time when she has begun pulling in the rope.

**We cover all the exercises in the chapter given below:-**

The solutions are explained in a student-friendly language and makes it easier for the learners.

**EXERCISE 6.1** - 3 Questions with Solutions in PDF

**EXERCISE 6.2** - 10 Questions with Solutions in PDF

**EXERCISE 6.3** - 16 Questions with Solutions in PDF

**EXERCISE 6.4** - 9 Questions with Solutions in PDF

**EXERCISE 6.5** - 17 Questions with Solutions in PDF

**EXERCISE 6.6 **- 10 Questions with Solutions in PDF

**Vedantu â€“ A Complete Guide for a Better Score in Boards!**

Vedantu offers solution sets for all subjects of Standard 10, as well as of other classes. You can avail our download option to get every set of answers, like those of Chapter 6 Maths Class 10 NCERT Solutions which can aid immensely in scoring better marks in your upcoming board examination. You can access our Maths NCERT Class 10 NCERT Solutions Class 10 Chapter 6 with easy online availability and free to download facilities.Â

You can download our NCERT Solutions for Class 10 Maths Chapter 6 PDF or access them from our app online whenever you are stuck with a question. Our teachers at Vedantu provide the most preferred methods of solving in our NCERT Class 10 Maths Chapter 6 solutions and others. The immense syllabus for your first Board examination can be handled with **CBSE NCERT Solutions **for Class 10 Chapter 6 Triangles to ensure that you achieve high scores.Â

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