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# NCERT Solutions for Class 9 Maths Chapter 7 - Triangles

Last updated date: 14th Sep 2024
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## Complete Resource of NCERT Solutions for Maths Chapter 7 Triangles Class 9 - Free PDF Download

NCERT Solutions for Triangles Class 9 focuses on triangles, an essential topic in geometry. This chapter helps students understand different kinds of triangles like isosceles, equilateral, and right-angled, and explores their properties and  an overview of triangle congruence and congruence principles. Learning about triangles is important because they are a basic shape used in many aspects of mathematics and everyday life. Also study characteristics and inequalities of triangles. The key areas covered in this chapter include understanding the angle sum property, which is that the angles inside any triangle add up to 180 degrees, and theorems like the Pythagorean theorem. Vedantu has provided clear explanations and step-by-step guidance to help students master these concepts. Concentrating on these areas is crucial for building a strong foundation in geometry.

Table of Content
1. Complete Resource of NCERT Solutions for Maths Chapter 7 Triangles Class 9 - Free PDF Download
2. Glance of NCERT Solutions for Class 9 Maths Trianglesย  Chapter 7 | Vedantu
3. Access Exercise Wise NCERT Solutions for Chapter 7 Maths Class 9
4. Exercises Under NCERT Solutions for Class 9 Maths Triangles Chapter 7
5. Access NCERT Answers for Class-9 Maths Chapter 7 โ Triangles
5.1Exercise- 7.1
5.2Exercise- 7.2
5.3Exercise- 7.3
6. Overview of Deleted Syllabus for CBSE Class 9 Maths Chapter 7 Triangles
7. Class 9 Maths Chapter 7: Exercises Breakdown
8. Other Study Material for CBSE Class 9 Maths Chapter 7
9. Chapter-Specific NCERT Solutions for Class 9 Maths
FAQs

## Glance of NCERT Solutions for Class 9 Maths Triangles  Chapter 7 | Vedantu

• Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

• Basic Proportionality Theorem (Thales Theorem) in a triangle, a line that is parallel to one side divides the other two sides proportionally.

• Angle Sum Property is the sum of angles in any triangle is always 180ยฐ.

• Congruence Criteria includes SAS, ASA, SSS, and RHS, which are essential for determining the congruence of triangles.

• This article contains detailed notes explaining the properties of triangles, different types of triangles, and criteria for congruence and similarity of triangles.

• Exercises are provided with solutions for applying congruence and similarity tests, proving properties using Pythagorasโ Theorem, and working with special types of triangles like equilateral and isosceles.

• This article contains links of chapter notes, important questions, exemplar solutions, exercises and video tutorials for better understanding.

• There are three exercises (21 fully solved questions) in Class 9th Maths Chapter 7 Fractions and Decimal.

## Access Exercise Wise NCERT Solutions for Chapter 7 Maths Class 9

 S.No. Current Syllabus Exercises of Class 9 Maths Chapter 7 1 NCERT Solutions of Class 9 Maths Triangles Exercise 7.1 2 3
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NCERT Solutions for Class 9 Maths Chapter 7 - Triangles
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## Exercises Under NCERT Solutions for Class 9 Maths Triangles Chapter 7

### Exercise 7.1:

This exercise comprises eight questions, and it mainly deals with the basic properties of triangles such as interior angles, exterior angles, and inequalities. The questions are mostly objective and require students to apply simple formulas to find the values of angles and sides of triangles.

### Exercise 7.2:

This exercise consists of eight questions and mainly focuses on the congruence of triangles. The questions in this exercise deal with the criteria for congruence of triangles such as SSS, SAS, ASA, RHS, and AAS. The questions require students to apply these criteria to determine if the given triangles are congruent or not.

### Exercise 7.3:

This exercise comprises five questions, and it deals with the properties of isosceles and equilateral triangles. The questions in this exercise require students to apply the properties of these triangles to find the values of angles and sides.

## Access NCERT Answers for Class-9 Maths Chapter 7 โ Triangles

### Exercise- 7.1

1. In quadrilateral ACBD, AC = AD and AB bisects $\angle A$ (See the given figure). Show that $\Delta ABC{\text{ }} \cong \Delta ABD$. What can you say about BC and BD?

Ans: Given: In quadrilateral ACBD, AC = AD and AB is bisected by $\angle A$

To find: Show that $\Delta ABC{\text{ }} \cong \Delta ABD$.

In $\Delta ABC{\text{ , }}\Delta ABD$

$AC{\text{ }} = {\text{ }}AD$ (Given)

$\angle CAB{\text{ }} = \angle DAB$     (AB bisects โ A)

$AB{\text{ }} = {\text{ }}AB$        (Common)

$\therefore \Delta ABC{\text{ }} \cong \Delta ABD$            (By SAS congruence rule)

$\therefore BC{\text{ }} = {\text{ }}BD$     (By CPCT)

Therefore, BC and BD are of equal lengths.

2. ABCD is a quadrilateral in which AD = BC and $\angle DAB{\text{ }} = \angle CBA$  (See the given figure). Prove that

(i)  $\Delta ABD{\text{ }} \cong \Delta BAC$

(ii) BD = AC

(iii)$\angle ABD{\text{ }} = \angle BAC$.

Ans : Given: ABCD is a quadrilateral where AD = BC and $\angle DAB{\text{ }} = \angle CBA$

To prove:

(i)  $\Delta ABD{\text{ }} \cong \Delta BAC$

(ii) BD = AC

(iii)$\angle ABD{\text{ }} = \angle BAC$.

In $\Delta ABD{\text{ , }}\Delta BAC$,

$\angle DAB{\text{ }} = \angle CBA$   (Given)

AB = BA     (Common)

$\therefore \Delta ABD{\text{ }} \cong \Delta BAC$              (By SAS congruence rule)

$\therefore BD{\text{ }} = {\text{ }}AC$          (By CPCT)

And, $\angle ABD{\text{ }} = \angle BAC$           (By CPCT)

3. AD and BC are equal perpendiculars to a line segment AB (See the given figure). Show that CD bisects AB.

Figure showing AD and BC equal perpendiculars to a line segment AB

Ans: Given: AD and BC are equal perpendiculars to a line segment AB

To prove: CD bisects AB.

In $\Delta BOC{\text{ , }}\Delta AOD$,

$\angle BOC{\text{ }} = \angle AOD$     (Vertically opposite angles)

$\angle CBO{\text{ }} = \angle DAO$        (Each right angle )

$\therefore \Delta BOC{\text{ }} \cong \Delta AOD$     (AAS congruence rule)

$\therefore BO{\text{ }} = {\text{ }}AO$     (By CPCT)

$\Rightarrow$CD bisects AB.

4. l and m are two parallel lines intersected by another pair of parallel lines p and q (see the given figure). Show that $\Delta ABC \cong \Delta CDA$.

Two  Parallel Lines L and M

Given: l and m are two parallel lines intersected by another pair of parallel lines p and q To prove:  $\Delta ABC \cong \Delta CDA$.

In $\Delta ABC{\text{ , }}\Delta CDA$,

$\angle BAC{\text{ }} = \angle DCA$       (Alternate interior angles, as$p{\text{ }}||{\text{ }}q$)

2AC = CA     (Common)

$\angle BCA{\text{ }} = \angle DAC$       (Alternate interior angles, as $l{\text{ }}||{\text{ }}m$)

$\therefore \Delta ABC \cong \Delta CDA$          (By ASA congruence rule)

## Overview of Deleted Syllabus for CBSE Class 9 Maths Chapter 7 Triangles

 Chapter Dropped Topics Triangles Exercise 7.6: Inequalities in triangles.

## Class 9 Maths Chapter 7: Exercises Breakdown

 Exercise Number of Questions Exercise 7.1 8 Questions (6 Short Answer Questions, 2 Long Answer Question) Exercise 7.2 8 Questions (6 Short Answer Questions, 2 Long Answer Question) Exercise 7.3 5 Questions (3 Short Answer Questions, 2 Long Answer Question)

## Conclusion

Triangle-related NCERT Solutions for Class 9 Chapter 7 from Vedantu are an excellent resource for understanding important geometric ideas. Because it provides the foundation for understanding more difficult geometrical structures that students will encounter in higher grades, this chapter is essential. Along with several congruence criteria including SAS, SSS, and ASA, it concentrates on key theorems such as the Pythagorean Theorem, Angle Sum Property, and Triangle Inequality. The importance of understanding both the theoretical and practical elements of these geometric principles is made clear by the three to five problems from this chapter that have frequently appeared in earlier year's question papers.

## Other Study Material for CBSE Class 9 Maths Chapter 7

 S.No. Important Links for Chapter 7 Triangles 1. Class 9 Triangles Chapter Notes 2. Class 9 Triangles Formula List 3. Class 9 Triangles Important Questions 4. Class 9 Triangles NCERT Exemplar 5. Class 9 Triangles RS Aggarwal Solutions for CBSE

## Chapter-Specific NCERT Solutions for Class 9 Maths

Given below are the chapter-wise NCERT Solutions for Class 9 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

## FAQs on NCERT Solutions for Class 9 Maths Chapter 7 - Triangles

1. What does class 9th maths chapter 7 include?

In class 9th maths chapter 7, students get to learn about triangles. In this chapter, the different properties of triangles are discussed, along with the meaning of triangles. The criteria for inequalities of triangles are also mentioned. The chapter ends with a discussion on the rules of congruence of triangles.

2. What are some important topics of Class 9 maths chapter 7 solutions?

Some important topics of class 9 maths chapter 7 are:

• Congruence of triangle

• Properties of triangle

• Inequalities of triangle

• Rules of congruence of triangle

3. From where can I download NCERT solutions class 9 maths chapter 7?

4. What are the properties of triangles?

There are six main properties of triangles. And these properties are:

• In a triangle, the angle opposite to the longer side is greater

• In a triangle, the side that is opposite to the greater angle is longer

• The sum of any two sides of a triangle is greater than the third side

• Every angle of an equilateral triangle measures to be 60 degrees

• The sides opposite to equal angles of a triangle are equal

• The angles opposite to equal sides of a triangle are equal

5. Are there any practice exercises available in NCERT Solutions for Chapter 7?

6. Is it essential to memorize the theorems and properties mentioned in this chapter?

While memorization can be helpful, it's more important to understand the underlying principles and how to apply them. Focus on grasping the concepts, and you'll find it easier to remember the theorems and properties.

7. Can I use NCERT Solutions as a primary study resource for Class 9 Mathematics Chapter 7?

NCERT Solutions are an excellent primary study resource, but it's also beneficial to refer to other textbooks, practice additional problems, and seek clarifications from teachers when needed to gain a more comprehensive understanding of the chapter.

8. Is triangles Class 9 difficult?

Class 9 triangles may seem difficult since they need an understanding of multiple fundamental geometry principles and theorems. However, the principles become easier to understand with thorough study and application. By providing clear explanations and visual aids, Vedantu's solutions and video tutorials may help in the simplification of these difficult concepts, making the learning experience easier and more enjoyable.

9. What is the basic concept of a triangle Class 9?

In Class 9 triangles, the basic concept of a triangle is to understand its characteristics, varieties, and angles and sides theorems. It covers the study of triangle congruence and similarity, the connection between a triangle's sides and angles, and the use of Pythagoras' Theorem in right-angled triangles.

10. What is the theory of triangles Class 9?

Triangle congruence and similarity criteria (like SSS, SAS, ASA, and RHS), Pythagoras' Theorem, properties like the idea that the sum of a triangle's angles is 180 degrees, and the triangle inequality theoremโwhich claims that the sum of any two of a triangle's sides is greater than the length of the third sideโare the main topics covered in Class 9's theory of triangles.

11. What is the median of a triangle in class 9 triangles?

A line segment from a vertex to the opposite side's midway is called a triangle's median. There are three medians in every triangle, and they are significant because they meet at a single location known as the centroid, which is the center of gravity of the triangle.

12. What are the 7 properties of a Triangles Class 9?

• The sum of the angles in a triangle is always 180 degrees.

• The exterior angle of a triangle is equal to the sum of the opposite interior angles.

• The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

• In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

• Each angle in an equilateral triangle is 60 degrees.

• The medians of a triangle intersect at a single point (centroid), which is the triangle's center of gravity.

• The perpendicular bisectors of a triangle's sides intersect at a point (circumcenter), which is equidistant from the triangleโs vertices.