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NCERT Solutions

Class 9

Maths

Chapter 6 Lines And Angles Exercise 6.1

NCERT Solutions for Class 9 Maths Chapter 6 Lines And Angles Ex 6.1

Q: 7. What are the important points to remember from the Chapter 6 Lines and Angles of Class 9 Maths?

The important points to remember are as below.A line segment is a line joining two endpoints.An angle more than 180 degrees is called the reflex angle.Complementary angles are the angles where the sum of the angles are 90 degrees.Intersecting lines cut each other always at 1 point.Parallel lines never intersect each other.The vertically opposite angles of two intersecting lines are equal.

Q: 8. Find the Interior angle If an exterior angle of a triangle is 105° and the two interior angles which are opposite are equal.

The exterior angle of a triangle is 105°.Let us assume the interior angle to be x.As the theorem, we know that Exterior angle=sum of the interior angle.x+x=105°2x=105°x=105/2=52.25°Each interior angle=52.25°Students need to be well versed with all the rules and the formulae of each topic so it becomes easy to solve based on the formulae. Make sure to revise all the formulae every day so it remains in the mind.

NCERT Solutions for Class 9 Maths Chapter 6 (Ex 6.1)

NCERT Solutions Class 9 Chapter 6 Exercise 6.1 PDF and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Maths Class 9 Exercise 6.1 Questions with Solutions to help you to revise complete Syllabus and Score More marks. Register and get all exercise solutions in your emails. Download Vedantu NCERT Solutions to get a better understanding of all the exercises questions. Make sure you download Maths NCERT Solution Class 9 curated by Master teachers at Vedantu.

Table of Content

Topics Covered in Class 9 Maths Chapter 6 Exercise 6.1

Definitions of line segment, ray, collinear points and non-collinear points, arms, vertex, and different types of angles.
Different ways of drawing two lines - (i) Intersecting lines (ii) Non-intersecting (parallel) lines.
Linear pair of angles, vertically opposite angles.

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Access Free PDF of NCERT Class 9 Maths Chapter 6: Lines and Angles - Exercise 6.1

Exercise No: 6.1

1. In the given figure, lines AB and CD intersect at O. If $∠ AOC+ ∠ BOE = 70^{\circ}$ and $∠ BOD = 4 0^{\circ}$ find $∠ BOE and reflex ∠ COE$ .

Ans:

Given: The line $AB$ and $CD$ intersect at $O$ .

$∠ AOC+ ∠ BOE = 70^{\circ}$

$∠ BOD = 4 0^{\circ}$ .

And $AB$ is a straight line; rays on the line are $OC$ and $OE$ .

If, $∠ AOC + ∠ COE + ∠ BOE = 18 0^{o}$

$\Rightarrow (∠ AOC + ∠ BOE) + ∠ COE = 18 0^{o}$

$\Rightarrow 7 0^{o} + ∠ COE = 18 0^{o}$

$\Rightarrow ∠ COE = 18 0^{o} - 7 0^{o} = 11 0^{o}$

$Reflex ∠ COE = 36 0^{o} -11 0^{o} =25 0^{o}$

$∴ Reflex ∠ COE =25 0^{o}$

Then $C D$ is a straight line; rays on the line are $O E$ and $O B$ .

And $∠ COE + ∠ BOE + ∠ BOD = 18 0^{o}$

$\Rightarrow 11 0^{o} + ∠ BOE+ 4 0^{o} = 18 0^{o}$

$\Rightarrow ∠ BOE = 18 0^{o} -15 0^{o} =3 0^{o}$

Hence, $∠ BOE = 3 0^{o}$ and $Reflex ∠ COE = 25 0^{o}$ .

2. In the given figure, lines XY and MN intersect at O. If $∠ POY=9 0^{o}$ and $a:b = 2:3$ , find c.

Ans:

Given: The line $XY$ and $MN$ intersect at $O$ .

Then, $∠ POY=9 0^{o}$ and $a:b = 2:3$ .

Let us assume the common ratio between a and b be x.

$∴ a = 2x,$ and $b = 3x$

And XY is a straight line, rays on the line are $O M$ and $O P$ .

$∵ ∠ XOM + ∠ MOP + ∠ POY = 18 0^{o}$

$b + a + ∠ POY = 18 0^{o}$

$3x + 2x + 9 0^{o} = 18 0^{o}$

$5x = 9 0^{o}$

$∴ x = 1 8^{o}$

Then,

$a = 2x = 2 \times 18 = 3 6^{o}$

$b = 3x = 3 \times 18 = 5 4^{o}$

Now, $M N$ is a straight line. Ray on the line is $O X$ .

The $Linear Pair$ is,

$b + c = 18 0^{o}$

$54^{o} + c = 18 0^{o}$

$c = 18 0^{o} - 54^{o}$

$= 126^{o}$

$∴ c = 126^{o}$

3. In the given figure, $∠ PQR = ∠ PRQ$ , then prove that $∠ PQS = ∠ PRT$ .

Ans:

In the given figure, ST is a straight line and ray QP stands on it.

The $Linear Pair$ is,

$∴ ∠ PQS + ∠ PQR = 18 0^{o}$

$∠ PQR = 18 0^{o} - ∠ PQS$ …… (1)

$∴ ∠ PRT + ∠ PRQ = 18 0^{o}$

$∠ PRQ = 18 0^{o} - ∠ PRT$ …… (2)

It is denoted as $∠ PQR = ∠ PRQ$ .

Now, equating equations (1) and (2). We get,

$18 0^{o} - ∠ PQS = 18 0^{o} - ∠ PRT$

$∴ ∠ PQS = ∠ PRT$

Hence proved.

4. In the given figure, if $x + y = w + z$ then prove that AOB is a line.

Ans:

Given: $x + y = w + z$ .

A complete angle becomes,

$x + y + z + w =36 0^{o}$

Then,

$x + y + x + y = 36 0^{\circ}$

$2 (x + y) = 36 0^{o}$

$∴ x + y = 18 0^{o}$

Since x and y form a linear pair, therefore, AOB is a line.

Hence proved.

5. In the given figure, $POQ$ is a line. Ray $OR$ is perpendicular to line $PQ$ . $OS$ is another ray lying between rays $OP$ and $OR$ . Prove that $∠ ROS = \frac{1}{2} (∠ QOS - ∠ POS)$

Ans:

Given: $OR ⊥ PQ$

$∠ POR = 9 0^{o}$

$∠ POS + ∠ SOR= 9 0^{o}$

$∠ ROS = 9 0^{o} + ∠ POS$ …… (1)

$∵ OR ⊥ PQ$ . Then,

$∠ QOR = 9 0^{o}$

$∠ QOS - ∠ ROS = 9 0^{o}$

$∠ ROS = ∠ QOS - 9 0^{o}$ …… (2)

Add equations (1) and (2), we obtain

$2 ∠ ROS = ∠ QOS - ∠ POS$

$∴ ∠ ROS = \frac{1}{2} (∠ QOS - ∠ POS)$

Hence proved.

6. It is given that $∠ XYZ = 6 4^{o}$ and $XY$ is produced to point $P$ . Draw a figure from the given information. If ray $YQ$ bisects $∠ ZYP$ , find $∠ XYQ$ and reflex $∠ QYP$ .

Ans:

Given: the line $Y Q$ bisects $∠ PYZ$ .

Hence, $∠ QYP = ∠ ZYQ$

It can be observed that $P X$ is a line. Rays on the line are $Y Q$ and $Y Z$ .

Then, $∠ XYZ + ∠ ZYQ + ∠ QYP = 18 0^{o}$

$64^{\circ} + 2 ∠ QYP = 180^{\circ}$

$2 ∠ QYP = 180^{\circ} - 64^{\circ} = 116^{\circ}$

$∠ QYP = 58^{\circ}$

$Also, ∠ ZYQ = ∠ QYP = 5 8^{o}$

$Reflex ∠ QYP = 36 0^{o} - 5 8^{o} = 302^{o}$

$∠ XYQ = ∠ XYZ + ∠ ZYQ$

$= 64^{\circ} + 58^{\circ} = 122^{\circ}$

$∴ ∠ XYQ = 122^{\circ}$ and $Reflex ∠ QYP = 302^{\circ}$ .

Conclusion:

NCERT Solutions for Class 9 Ex 6.1 by Vedantu is a helpful resource that simplifies the understanding of basic geometry concepts. This Exercise focuses on fundamental definitions of Line, line-segment, angle, ray and different types of angles, pairs of angles. The important concepts include supplementary angles, complementary angles, vertically opposite angles and properties of angles. Vedantu’s detailed solutions and clear explanations ensure students can easily follow along and strengthen their foundational knowledge in geometry, which is crucial for tackling more complex topics in later exercises.

CBSE Class 9 Maths Chapter 6 Other Study Materials

S.No.	Important Links for Chapter 6 Lines and Angles
1	Class 9 Lines and Angles Revision Notes
2	Class 9 Lines and Angles Important Questions
3	Class 9 Lines and Angles Formula
4	Class 9 Lines and Angles NCERT Exemplar Solution
5	Class 9 Lines and Angles RD Sharma Solutions
6	Class 9 Lines and Angles RS Aggarwal Solutions

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.

S.No.	NCERT Solutions Class 9 Maths Chapter-wise List
1	Chapter 1 - Number Systems
2	Chapter 2 - Polynomials
3	Chapter 3 - Coordinate Geometry
4	Chapter 4 - Linear Equations in Two Variables
5	Chapter 5 - Introduction to Euclids Geometry
6	Chapter 6 - Lines and Angles
7	Chapter 7 - Triangles
8	Chapter 8 - Quadrilaterals
9	Chapter 9 - Circles
10	Chapter 10 - Heron's Formula
11	Chapter 11 - Surface Areas and Volumes
12	Chapter 12 - Statistics

Important Study Materials for CBSE Class 9 Maths

S.No.	Important Study Material for Maths Class 9
1	CBSE Class 9 Maths NCERT Books
2	CBSE Class 9 Maths Revision Notes
3	CBSE Class 9 Maths Important Questions
4	CBSE Class 9 Maths NCERT Solutions
5	CBSE Class 9 Maths NCERT Exemplar Solution
6	CBSE Class 9 Maths RS Aggarwal Solutions
7	CBSE Class 9 Maths RD Sharma Solutions
8	CBSE Class 9 Maths Formulas
9	CBSE Class 9 Maths Syllabus
10	CBSE Class 9 Maths Sample Papers

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FAQs on NCERT Solutions for Class 9 Maths Chapter 6 Lines And Angles Ex 6.1

1. What are the different types of angles?

There are types of angles - acute angle, right angle, obtuse angle, straight angle, and reflex angle.

An acute angle comes in between 0° and 90° while a right angle is equal to 90°. An obtuse angle is defined as an angle which is greater than 90° but less than 180°. A straight angle is exactly equal to 180°. A reflex angle is greater than 180° but less than 360°. However, if the sum of two angles is 90°, then they are called complementary angles. And, if two angles’ total sum is 180°, then they are called supplementary angles.

2. How can I download the NCERT Solutions for Class 9 Maths Chapter 6 Exercise 6.1?

You can directly download NCERT Solutions directly from our website. You can also download our app from the google play store to easily access the solutions PDF for better exam preparation.

3. What are NCERT Solutions for Class 9 Maths Chapter 6 Exercise 6.1 all about?

NCERT Solutions for Class 9 Maths Chapter 6 Exercise 6.1 will give a brief idea to the students about the various types of angles (acute, obtuse, right, straight, reflex, complementary and supplementary).

The new concepts in the exercise depict the Intersecting and non-intersecting lines, and then Pairs of Angles are elaborately explained in detail along with a theorem-proof and some examples in the NCERT textbook.

The questions in the exercise are based on finding the value of angles from the given diagrams. In the Class 9 Maths Solutions, we have put our best foot forward to provide the step by step solution along with the diagrams for the convenience of the students. The questions are also solved by the expert teachers in the easiest possible method, so the student can easily understand it and grasp all the concepts for a better understanding.

4. Is it beneficial to get access to the NCERT Solutions for Class 9 Maths Chapter 6 Exercise 6.1?

NCERT Solution for Maths Chapter 6 Lines and Angles Exercise 6.1 is undoubtedly beneficial for all Class 9 students. There are step-by-step solutions are provided. Experts who have created these solutions have also provided alternative methods to solve a problem (wherever possible). All the methods and solutions are easy to understand by students in every category. Hence you should download it from your website or mobile app at absolutely no cost. So that you can study at your own pace.

5. Which topic is Exercise 6.1 of Chapter 6 of Class 9 Maths based on?

Chapter 6 of Class 9 Maths is Lines and Angles. Exercise 6.1, the first exercise, deals with sums related to angles. This is a very important exercise and hence, the questions must be solved with care and focus. The calculation and the measurement should be accurate. Practising every day is the best solution to be perfect in the chapter. You can also refer to NCERT Solutions from Vedantu for comprehensive preparation and understanding of the chapter.

6. Is Exercise 6.1 of Chapter 6 of Class 9 Maths difficult?

Nothing is difficult or impossible but it always depends on the way you learn, practice and focus. Exercise 6.1 is not difficult, but it is indeed important. NCERT Solutions by Vedantu can prove to be of great help as it provides students with different methods of solving the problems, and also allows them to gain confidence by clearing all concepts thoroughly. The solutions are in easy and simple steps where all the students can understand clearly.

These solutions are available on Vedantu's official website(vedantu.com) and mobile app free of cost.

7. What are the important points to remember from the Chapter 6 Lines and Angles of Class 9 Maths?

The important points to remember are as below.

A line segment is a line joining two endpoints.
An angle more than 180 degrees is called the reflex angle.
Complementary angles are the angles where the sum of the angles are 90 degrees.
Intersecting lines cut each other always at 1 point.
Parallel lines never intersect each other.
The vertically opposite angles of two intersecting lines are equal.

8. Find the Interior angle If an exterior angle of a triangle is 105° and the two interior angles which are opposite are equal.

The exterior angle of a triangle is 105°.

Let us assume the interior angle to be x.

As the theorem, we know that

Exterior angle=sum of the interior angle.

x+x=105°

2x=105°

x=105/2=52.25°

Each interior angle=52.25°

Students need to be well versed with all the rules and the formulae of each topic so it becomes easy to solve based on the formulae. Make sure to revise all the formulae every day so it remains in the mind.

9. What is the type of triangle if one of the angles is equal to the sum of the other two angles?

If one of the angles of a triangle is equal to the sum of the other two angles then the triangle is right-angled. As we know that the sum of the interior angles of a triangle is 180° and one of the angles of a triangle is 90° the of the other two angles is 90° which are acute angles. Furthermore, students can practise extra questions so they get the right concept of the chapter.

10. Why is it important to understand the properties of angles in Class 9 Maths Ex 6.1?

Understanding the properties of angles is crucial for solving geometric problems, proving theorems, and understanding the relationships between different geometric shapes.

11. How can practicing Class 9 Maths Ch 6 Ex 6.1 help in exams?

Practicing Exercise 6.1 helps students understand the foundational concepts of lines and angles, which are frequently tested in exams. Regular practice ensures familiarity with problem-solving techniques and improves accuracy.

NCERT Solutions for Class 9 Maths Chapter 6 Lines And Angles Ex 6.1

NCERT Solutions for Class 9 Maths Chapter 6 (Ex 6.1)

Topics Covered in Class 9 Maths Chapter 6 Exercise 6.1

Access Free PDF of NCERT Class 9 Maths Chapter 6: Lines and Angles - Exercise 6.1

Exercise No: 6.1

Conclusion:

CBSE Class 9 Maths Chapter 6 Other Study Materials

Chapter-Specific NCERT Solutions for Class 9 Maths

Important Study Materials for CBSE Class 9 Maths

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FAQs on NCERT Solutions for Class 9 Maths Chapter 6 Lines And Angles Ex 6.1