## RD Sharma Solutions Class 9 Maths Chapter 9 – Free PDF Download

**RD Sharma Class 9 Solutions** are provided here for students to clear all their doubts regarding triangles, as well as solve problems based on this topic. The solutions are helpful for students to practise on a regular basis and secure high marks in their exams. RD Sharma Solutions Class 9 Chapter 9 – Triangle and Its Angles, includes three exercises, and all questions are solved by BYJU’S subject experts. In this chapter, students will study a plane figure formed by three non-parallel lines in a plane called a triangle. A triangle is a 2D geometrical figure consisting of three edges and three vertices.

The three angles are made by the three sides of a triangle, and the sum of all the internal angles of a triangle is equal to 180 degrees. Triangles are categorised based on their angles: Acute Triangle, Right Angle and Obtuse Triangle. For a better understanding of concepts, students can download the solutions in PDF whenever required. RD Sharma Solutions are the best materials for students to become proficient in Mathematics. The solutions are designed based on the latest **CBSE **syllabus as per the student’s intelligence quotient.

## RD Sharma Solutions for Class 9 Maths Chapter 9 Triangle and Its Angles

### Access Answers to Maths RD Sharma Solutions for Class 9 Chapter 9 Triangle and Its Angles

Exercise 9.1 Page No: 9.9

**Question 1: In a ΔABC, if ∠A = 55 ^{0}, ∠B = 40^{0}, find ∠C.**

**Solution:**

Given: ∠A = 55^{0}, ∠B = 40^{0}

We know, sum of all angles of a triangle is 180^{0}

∠A + ∠B + ∠C = 180^{0}

55^{0 }+ 40^{0 }+ ∠C=180^{0}

95^{0 }+ ∠C = 180^{0}

∠C = 180^{0} − 95^{0}

*∠C = 85 ^{0}*

**Question 2: If the angles of a triangle are in the ratio 1:2:3, determine three angles.**

**Solution:**

Angles of a triangle are in the ratio 1:2:3 (Given)

Let the angles be x, 2x, 3x

Sum of all angles of triangles = 180^{0}

x + 2x + 3x = 180^{0}

6x = 180^{0}

x = 180^{0}/6

x = 30^{0}

Answer:

*x = 30 ^{0}*

2x = 2(30)^{0} *= 60 ^{0}*

3x = 3(30)^{ 0} = *90 ^{0}*

**Question 3: The angles of a triangle are (x − 40) ^{0}, (x − 20)^{ 0 }and (1/2 x − 10)^{ 0}. Find the value of x.**

**Solution:**

The angles of a triangle are (x − 40)^{0}, (x − 20)^{ 0 }and (1/2 x − 10)^{ 0}

Sum of all angles of triangle = 180^{0}

(x − 40)^{0} + (x − 20)^{ 0 }+ (1/2 x − 10)^{ 0} = 180^{0}

5/2 x – 70^{0} = 1800

5/2 x = 180^{0} + 70^{0}

5x = 2(250)^{ 0}

x = 500^{0}/5

*x = 100 ^{0}*

**Question 4: The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angles is 10 ^{0}, find the three angles.**

**Solution:**

The difference between two consecutive angles is 10^{0} (given)

Let x, x + 10^{0}, x + 20^{0} be the consecutive angles

x + x + 10^{0} + x + 20^{0} = 180^{0}

3x + 30^{0} = 180^{0}

3x = 180^{0}– 30^{0}

3x = 150^{0}

or *x = 50 ^{0}*

Again,

x + 10^{0} = 50^{0} + 10^{0} *= 60 ^{0}*

x+20^{0} = 50^{0} + 20^{0} *= 70 ^{0}*

Answer: *Three angles are 50 ^{0},60^{0} and 70^{0}.*

**Question 5: Two angles of a triangle are equal and the third angle is greater than each of those angles by 30 ^{0}. Determine all the angles of the triangle.**

**Solution:**

Two angles of a triangle are equal and the third angle is greater than each of those angles by 30^{0}. (Given)

Let x, x, x + 30^{0} be the angles of a triangle.

Sum of all angles in a triangle = 180^{0}

x + x + x + 30^{0} = 180^{0}

3x + 30^{0} = 180^{0}

3x = 150^{0}

or x = 50^{0}

And x + 30^{0} = 50^{0} + 30^{0} = 80^{0}

Answer: *Three angles are 50 ^{0}, 50^{0} and 80^{0}.*

**Question 6: If one angle of a triangle is equal to the sum of the other two, show that the triangle is a right angle triangle.**

**Solution:**

One angle of a triangle is equal to the sum of the other two angles (given)

To Prove: One of the angles is 90^{0}

Let x, y and z are three angles of a triangle, where

z = x + y …(1)

Sum of all angles of a triangle = 180^{0}

x + y + z = 180^{0}

z + z = 180^{0} (Using equation (1))

2z = 180^{0}

z = 90^{0} (Proved)

*Therefore, triangle is a right angled triangle.*

Exercise 9.2 Page No: 9.18

**Question 1: The exterior angles, obtained on producing the base of a triangle both ways are 104 ^{0} and 136^{0}. Find all the angles of the triangle.**

**Solution:**

∠ACD = ∠ABC + ∠BAC [Exterior angle property]

Find ∠ABC:

∠ABC + ∠ABE = 180^{0 } [Linear pair]

∠ABC + 136^{0 }= 180^{0}

∠ABC = 44^{0}

Find ∠ACB:

∠ACB + ∠ACD = 180^{0} [Linear pair]

∠ACB + 104^{0} = 180^{0}

∠ACB = 76^{0}

Now,

Sum of all angles of a triangle = 180^{0}

∠A + 44^{0} + 76^{0} = 180^{0}

∠A = 180^{0 }− 44^{0 }−76^{0}

∠ A = 60^{0}

Answer: *Angles of a triangle are ∠ A = 60 ^{0}, ∠B = 44^{0} and ∠C = 76^{0}*

**Question 2: In a △ABC, the internal bisectors of ∠B and ∠C meet at P and the external bisectors of ∠B and ∠C meet at Q. Prove that ∠BPC + ∠BQC = 180 ^{0}.**

**Solution**:

In triangle ABC,

BP and CP are internal bisector of ∠B and ∠C respectively

=> External ∠B = 180^{o} – ∠B

BQ and CQ are external bisector of ∠B and ∠C respectively.

=> External ∠C = 180^{ o} – ∠C

In triangle BPC,

∠BPC + 1/2∠B + 1/2∠C = 180^{o}

∠BPC = 180^{ o} – 1/2(∠B + ∠C) …. (1)

In triangle BQC,

∠BQC + 1/2(180^{ o} – ∠B) + 1/2(180^{ o} – ∠C) = 180^{ o}

∠BQC + 180^{ o} – 1/2(∠B + ∠C) = 180^{ o}

*∠BPC + ∠BQC = 180 ^{ o} [Using (1)]*

Hence Proved.

**Question 3: In figure, the sides BC, CA and AB of a △ABC have been produced to D, E and F respectively. If ∠ACD = 105 ^{0} and ∠EAF = 45^{0}, find all the angles of the △ABC.**

**Solution:**

∠BAC = ∠EAF = 45^{0} [Vertically opposite angles]

∠ACD = 180^{0} – 105^{0} = 75^{0} [Linear pair]

*∠ABC = 105 ^{0} – 45^{0} = 60^{0} [Exterior angle property]*

**Question 4: Compute the value of x in each of the following figures:**

**(i)**

**Solution:**

∠BAC = 180^{0} – 120^{0} = 60^{0} [Linear pair]

∠ACB = 180^{0} – 112^{0} = 68^{0} [Linear pair]

Sum of all angles of a triangle = 180^{0}

x = 180^{0 }− ∠BAC − ∠ACB

= 180^{0 }− 60^{0 }− 68^{0 }= 52^{0}

Answer: *x = 52 ^{0}*

**(ii)**

**Solution:**

∠ABC = 180^{0} – 120^{0} = 60^{0} [Linear pair]

∠ACB = 180^{0} – 110^{0} = 70^{0} [Linear pair]

Sum of all angles of a triangle = 180^{0}

x = ∠BAC = 180^{0} − ∠ABC − ∠ACB

= 180^{0} – 60^{0} – 70^{0} = 50^{0}

Answer: *x = 50 ^{0}*

**(iii)**

**Solution:**

∠BAE = ∠EDC = 52^{0} [Alternate angles]

Sum of all angles of a triangle = 180^{0}

x = 180^{0} – 40^{0} – 52^{0} = 180^{0 }− 92^{0} = 88^{0}

Answer: *x = 88 ^{0}*

**(iv) **

**Solution:**

CD is produced to meet AB at E.

∠BEC = 180^{0} – 45^{0} – 50^{0} = 85^{0} [Sum of all angles of a triangle = 180^{0}]

∠AEC = 180^{0} – 85^{0} = 95^{0} [Linear Pair]

Now, x = 95^{0} + 35^{0} = 130^{0} [Exterior angle Property]

Answer: *x = 130 ^{0}*

**Question 5: In figure, AB divides ∠DAC in the ratio 1 : 3 and AB = DB. Determine the value of x.**

**Solution**:

$∠DAC=180°−108°=72°$

$BAC/∠DAB =1/3$

Exercise VSAQs Page No: 9.21

**Question 1: Define a triangle.**

**Solution:** *Triangle is a three-sided polygon that consists of three edges and three vertices. The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees.*

**Question 2: Write the sum of the angles of an obtuse triangle.**

**Solution:** *The sum of angles of obtuse triangle = 180°.*

**Question 3: In △ABC, if ∠B = 60**^{0}**, ∠C = 80**^{0}** and the bisectors of angles ∠ABC and ∠ACB meet at point O, then find the measure of ∠BOC.**

**Solution**:

∠B = 60^{0}, ∠C = 80^{0} (given)

As per question:

∠OBC = 60^{0}/2 = 30^{0} and

∠OCB = 80^{0}/2 = 40^{0}

In triangle BOC,

∠OBC + ∠OCB + ∠BOC = 180 ^{0}

^{0}]

30^{0} + 40^{0} + ∠BOC = 180^{0}

*∠BOC = 110 ^{0}*

**Question 4: If the angles of a triangle are in the ratio 2:1:3, then find the measure of smallest angle.**

**Solution**:

Let angles of a triangles are 2x, x and 3x, where x is the smallest angle.

To find: measure of x.

As, Sum of angles of a triangle = 180^{0}

2x + x + 3x = 180^{0}

6x = 180^{0}

*x = 30 ^{0}*. Answer

**Question 5: If the angles A, B and C of △ABC satisfy the relation B – A = C – B, then find the measure of ∠B.**

**Solution: **

Sum of angles of a triangle = 180^{0}

A + B + C = 180^{0} …(1)

B – A = C – B …(Given)

2B = C + A …(2)

(1) => 2B + B = 180^{0}

3B =180^{0}

Or *B = 60 ^{0}*

## RD Sharma Solutions for Class 9 Maths Chapter 9 Triangle and Its Angles

In the 9th Chapter of Class 9 RD Sharma Solutions, students will study important concepts listed below.

- Triangle introduction
- Types of triangles
- Some important theorems on triangles

Students can download all solutions listed in the RD Sharma Class 9 textbook and start practising to score good marks.

## Frequently Asked Questions on RD Sharma Solutions for Class 9 Maths Chapter 9

### Why should we download RD Sharma Solutions for Class 9 Maths Chapter 9 to learn concepts better?

RD Sharma Solutions for Class 9 Maths are described in a precise manner by BYJU’S experts. The solutions are well-structured in a simple and understandable language to improve the grasping abilities of students. In order to gain proficiency in Maths, students must practise RD Sharma Solutions as many times as possible. They also can make use of solutions in PDF to improve the skills required for solving problems effortlessly. The solutions can be accessed anytime and anywhere as per their needs.

### Why are RD Sharma Solutions for Class 9 Maths Chapter 9 essential to score high marks in the final exams?

RD Sharma Solutions are the best sources to understand complex concepts in an easy manner. Students who follow these solutions diligently clear their doubts quickly and also enhance their problem-solving skills. RD Sharma Solutions for Class 9 Maths Chapter 9 are well formulated by subject experts at BYJU’S in a step-by-step manner. This allows the students to boost their confidence to solve any type of problem with ease and score good marks in the final examination.

### Give a brief summary of concepts present in RD Sharma Solutions for Class 9 Maths Chapter 9.

The concepts discussed in RD Sharma Solutions for Class 9 Maths Chapter 9 are as follows:

Triangle introduction

Types of triangles

Some important theorems on triangles

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