NCERT CLASS 6TH MATHS CHAPTER 2: LINES AND ANGLES

Understand NCERT Class 6 Maths Chapter 2: Lines and Angles with easy explanations, examples, and exercises. Learn key concepts like types of angles, intersecting lines, and properties of shapes. Perfect for CBSE students!"

Here, we will explore some of the most basic ideas of geometry including points, line segment, rays, lines and angles.These ideas form the building blocks of ‘plane geometry’, and will help us in understanding more advanced topics in geometry such as the construction and analysis of different shapes.

• NCERT CLASS 6TH MATHS CHAPTER 1 PATTERNS IN MATHEMATICS
• NCERT CLASS 6TH MATHS CHAPTER 3 NUMBER PLAY 
• NCERT CLASS 6TH MATHS CHAPTER 4 DATA HANDLING AND PRESENTATION 
• NCERT CLASS 6TH MATHS CHAPTER 5 PRIME TIME 
• NCERT CLASS 6TH MATHS CHAPTER 6 PERIMETER AND AREA
• NCERT CLASS 6TH MATHS CHAPTER 7 FRACTION 
• NCERT CLASS 6TH MATHS CHAPTER 8 PLATING WITH CONSTRUCTIONS
• NCERT CLASS 6TH MATHS CHAPTER 9 SYMMETRY
• NCERT CLASS 6TH MATHS CHAPTER 10 THE OTHER SIDE OF ZERO

Topic 2.1 points

Topic 2.2 Line Segment

Topic 2.3 Line     

Topic 2.1 Ray

 NCERT CLASS 6TH MATHS CHAPTER 2 LINES AND ANGLES

        

Figure It Out

Question 1

Can you help Rihan and Sheetal find their answers?

Solution:

Rihan can draw infinite number of lines that pass through the point.

Sheetal can draw only onle line that pass through two given points.

Question 2
Name the line segments in the given figure. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments?

Solution:

Five line segments are LM, MP, PQ QR. 

Point M is common point of line segments LM and MP.

Point P is common point of line segments MP and PQ.

Point Q is common point of line segments PQ and QR

Question 3

Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays?

 NCERT CLASS 6TH MATHS CHAPTER 2 LINES AND ANGLES

Solution:

Name of rays: Ray TA, Ray TN, Ray TB, Ray NB. 

No, T is not the starting point of each ray because Ray NB starts from point N.

Question 4

Draw a rough figure and write labels appropriately to illustrate each of the following:

Solution:

For (a):

 

For (b)

For (c)

For (d)

Question 5
In the given figure, name:
a. Five points
b. A line
c. Four rays
d. Five line segments

Solution:

 NCERT CLASS 6TH MATHS CHAPTER 2 LINES AND ANGLES

Question 6
Here is a ray OA . It starts at O and passes through the point A. It also passes through the point B.
a. Can you also name it as ray OB? Why? 
b. Can we write ray OAas ray AO? Why or why not?

Solution (a):

Yes,we can also name it as ray OB, because point B is on the ray OA and direction of ray OB is same as ray OA. 

Solution (b);

NO, we cannot write  ray OA as ray AO, because by doing this the direction will change which is not correct.

Topic 2.5 Angles

When two rays start from a common point they formed an angle.

Figure It Out

Question 1
Can you find the angles in the given pictures? Draw the rays forming any one of the angles and name the vertex of the angle.

Solution:

 NCERT CLASS 6TH MATHS CHAPTER 2 LINES AND ANGLES

Question 2
Draw and label an angle with arms ST and SR.

Slution:

Question 3
Explain why ∠APC cannot be labelled as ∠P.

Solution:

Here, ray PA, ray PB and ray PC are starting from common point P and making three different angles ∠APB, ∠BPC and ∠APC at P. Point P is common point for all the three angles therefore we cannot be labelled ∠APC as ∠P.

Question 4
Name the angles marked in the given figure.

Solution: 

Marked angles are ∠PTR and ∠QTR

Question 5

Mark any three points on your paper that are not on one line. Label them A, B, C. Draw all possible lines going through pairs of these points. How many lines do you get? Name them. How many angles can you name using A, B, C? Write them down, and mark each of them with a curve.

Solution:

 NCERT CLASS 6TH MATHS CHAPTER 2 LINES AND ANGLES

We get three lines line AB, line BC and line CA.We also get three angles ∠ABC, ∠BCA and ∠CAB.

Question 6

Now mark any four points on your paper so that no three of them are on one line. Label them A, B, C, D. Draw all possible lines going through pairs of these points. How many lines do you get? Name them. How many angles can you name using A, B, C, D? Write them all down, and mark each of them with a curve.

Solution:

 We get six lines line AB, line BC, line CD, line DA, line AC and line BD .We also get 12 angles ∠ABC, ∠BCD, ∠CDA, ∠DAB, ∠ABD, ∠DBC, ∠BCA, ∠ACD, ∠CDB, ∠BDA, ∠DAC,and ∠CAB.

Topic 2.6 Comparing Angles

Figure It Out

Question 1
Fold a rectangular sheet of paper, then draw a line along the fold created. Name and compare the angles formed between the fold and the sides of the paper. Make different angles by folding a rectangular sheet of paper and compare the angles. Which is the largest and smallest angle you made?

Solution:

Activity based question do by yourself

Question 2
In each case, determine which angle is greater and why.
a. ∠AOB or ∠XOY
b. ∠AOB or ∠XOB
c. ∠XOB or ∠XOC

Solution:

For (a)

 In ∠AOB and ∠XOY rotation of ray OC to ray OA is grater than rotation of ray OY to ray OX. Therefore ∠AOB is greater ∠XOY.

For (b)

 In ∠AOB and ∠XOB rotation of ray OC to ray OA is grater than rotation of ray OB to ray OY. Therefore ∠AOB is greater ∠XOB.

For (c)

 In ∠XOB and ∠XOC rotation of ray OB to ray OX is equal to rotation of ray OC to ray OX. Therefore ∠XOB is equal to ∠XOC.

Question 3
Which angle is greater: ∠XOY or ∠AOB? Give reasons.

Solution:

Here, ∠XOY is greater than ∠AOB because in both angles rotation of ray OY to ray OX is greater than rotation of ray OB to ray OA.

Topic 2.7 Making Rotating Arms
Topic 2.8 Special Types of Angles

Figure It Out

Question 1
How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?

Solution:

In our classroom window covers four right angles. 

Yes, we see other right angles in our class like corner of class room,edges of black board and class room door also contain right angles. 

Question 2
Join A to other grid points in the figure by a straight line to get a straight angle. What are all the different ways of doing it?

 NCERT CLASS 6TH MATHS CHAPTER 2 LINES AND ANGLES

Solution:

Question 3
Now join A to other grid points in the figure by a straight line to get a right angle. What are all the different ways of doing it?

 NCERT CLASS 6TH MATHS CHAPTER 2 LINES AND ANGLES

Solution:

Question 4
Get a slanting crease on the paper. Now, try to get another crease that is perpendicular to the slanting crease.
a. How many right angles do you have now? Justify why the angles are exact right angles.
b. Describe how you folded the paper so that any other person who doesn’t know the process can simply follow your description to get the right angle.

Solution

For (a)

We have four right angles.All are exact right angle because by intersecting of both lines we get four equal quadrant and and each quadrant has right angle.

For (b) 

First we take a piece of paper and make a slant fold after fold again from its folded edge. 

Figure It Out

Question 1
Identify acute, right, obtuse and straight angles in the previous figures.

Solution:

Question 2
Make a few acute angles and a few obtuse angles. Draw them in different orientations.

Solution:

NCERT CLASS 6TH MATHS CHAPTER 2 LINES AND ANGLES

Question 3
Do you know what the words acute and obtuse mean? Acute means sharp and obtuse means blunt. Why do you think these words have been chosen?

Answer:

This word is used because when an acute angle is formed then it has a sharp vertex but when an obtuse angle is formed then the vertex is not sharp. Therefore these word have been chosen.

Qusetion 4
Find out the number of acute angles in each of the figures below.

What will be the next figure and how many acute angles will it have?Do you notice any pattern in the numbers?

Solution:

Here,number of acute angles in next triangle increase by 9 from previous triangle.

3 

3 + 9 = 12

12 + 9 = 21

21 + 9 =30 and so on.

Topic 2.9 Measuring Angles

Figure It Out

Question 1
Find the degree measures of the following angles using your protractor.

Solution: 

Question 2
Find the degree measures of different angles in your classroom using your protractor.

Solution:

Angle between two edges of wall = 90°

Angles between adjacent sides of black board =90°

Angle between two adjacent sides of equilateral triangle =60°

etc.

Question 3
Find the degree measures for the angles given below. Check if your paper protractor can be used here!

Solution:

Question 4
How can you find the degree measure of the angle given below using a  ?

Solution:

First we find the measure of opposite angle of marked angle then we subtract obtained measure of  opposite angle from 360° after that we will get marked angle.

opposite angle of marked angle = 100°

Marked angle = 360° - 100°  = 260°

Question 5
Measure and write the degree measures for each of the following angles:

Solution:

Question 6
Find the degree measures of ∠BXE, ∠CXE, ∠AXB and ∠BXC.

Solution:

m∠BXE = 115°

m∠CXE = 85°

m∠AXB = 65°

m∠BXC = m∠BXE - m∠CXE = 115° - 85° = 30°

Question 7
Find the degree measures of ∠PQR, ∠PQS and ∠PQT.

Solution:

m∠PQR =45°

m∠PQS =110°

m∠PQT =150°

Question 8
Make the paper craft as per the given instructions. Then, unfold and open the paper fully. Draw lines on the creases made and measure the angles formed.

Solution: Activity based question fold and do by yourself.

Question 9
Measure all three angles of the triangle shown in the given figure (a), and write the measures down near the respective angles. Now add up the three measures. What do you get? Do the same for the triangles in the given figure  (b) and (c). Try it for other triangles as well, and then make a conjecture for what happens in general! We will come back to why this happens in a later year.

Solution:

(a)45°+65°+70° = 180°

(b)60°+64°+56° = 180°

(c) 96°+52°+32° = 180°

When we add all angles of the given triange (a) we get sum as 180° same work when we repeated for triangle (b) and triangle (c) we also get 180°. 

By this sum of angles of triangle we can say that sum of all angles of triangle is 180°.

Figure It Out

Question 1
Angles in a clock:
a. The hands of a clock make different angles at different times. At 1 o’clock, the angle between the hands is 30°. Why?
b. What will be the angle at 2 no’clock? And at 4 o’clock? 6 o’clock?c. Explore other angles made by the hands of a clock.

Solution: 

For (a)

Complete angle =360°

Total equal part in clock (1 to 12) = 12

Angle between to consecutive number in the clock = 360° ÷ 12 = 30°

For (b)

angle at 2 no’clock = 60°

angle at 4 no’clock = 120°

angle at 6 no’clock  = 180°

Question 2
The angle of a door: Is it possible to express the amount by which a door is opened using an angle? What will be the vertex of the angle and what will be the arms of the angle?

Solution:

Yes, it possible to express the amount by which a door is opened using an angle.

Question 3
Vidya is enjoying her time on the swing. She notices that the greater the angle with which she starts the swinging, the greater is the speed she achieves on her swing. But where is the angle? Are you able to see any angle?

Solution:

Vertical line show the initial position of Vidhya and she will swing around this line.

Question 4
Here is a toy with slanting slabs attached to its sides; the greater the angles or slopes of
the slabs, the faster the balls roll. Can angles be used to describe the slopes of the slabs?
What are the arms of each angle? Which arm is visible and which is not?

Solution:

Yes we can use angles to describe the slopes of the slabs.

horizontal arm are not visible and slanting arms are visible.

Question 5
Observe the images below where there is an insect and its rotated version. Can angles be used to describe the amount of rotation? How? What will be the arms of the angle and the vertex?

Hint: Observe the horizontal line touching the insects.

Solution:

Yes we can use angles to discribe the amount of rotation insect 1st rotate left by 90° and insect 2nd rotate by right 90°.

Topic 2.10 Drawing Angles

Figure It Out

Question 1
In the given figure, list all the angles possible. Did you find them all? Now, guess the measures of all the angles. Then, measure the angles with a protractor. Record all your numbers in a table. See how close your guesses are to the actual measures.

Solution:

Angle’s Name	Guess (measure)	Actual measure
∠1	80˚	73˚
∠2	100˚	103˚
∠3	95˚	80˚
∠4	105˚	100˚
∠5	100˚	100˚
∠6	80˚	80˚
∠7	100˚	103˚
∠8	80˚	73˚
∠9	105˚	100˚
∠10	75˚	80˚
∠11	80˚	80˚
∠12	100˚	100˚

Question 2
Use a protractor to draw angles having the following degree measures:
a. 110°         b. 40°         c. 75°         d. 112°         e. 134°

Solution:

Question 3

Draw an angle whose degree measure is the same as the angle given below:

Also, write down the steps you followed to draw the angle.

Solution:

Put a paper below the given angle and trace it so you will get creases of given angle on the paper then dark the creases by pencil and measure obtained angle you get obtained angle m∠ABC is equal to given angle m∠IHJ.

Hence,m∠ABC = m∠IHJ =115°

Topic 2.11 Types of Angles and their Measures

LINES AND ANGLES CLASS 6

Figure It Out

Question 1
In each of the below grids, join A to other grid points in the figure by a straight line to get:
a. An acute angle

Solution:

b. An obtuse angle

Solution:

c. A reflex angle

Solution:

Question 2
Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. a.∠PTR     b. ∠PTQ     c. ∠PTW     d. ∠WTP

Solution:

(a) ∠PTR   = 30° (acute angle)

(b) ∠PTQ  = 60° (acute angle)

(c) ∠PTW  = 103° (obtuse angle)

(d)∠WTP   = 257° (reflex angle)

Figure It Out

Question 1
Draw angles with the following degree measures:
a. 140°     b. 82°     c. 195°     d. 70°     e. 35°

Solution:

Question 2
Estimate the size of each angle and then measure it with a protractor:

Classify these angles as acute, right, obtuse or reflex angles.

Solution:

Angle’s Name	Estimate measure	Actual measure	Type
a	40˚	45˚	Acute angle
b	170˚	175˚	Obtuse angle
c	120˚	120˚	Obtuse angle
d	30˚	30˚	Acute angle
e	100˚	95˚	Obtuse angle
f	350˚	350˚	Reflex angle

Question 3
Make any figure with three acute angles, one right angle and two obtuse angles.

Solution:

3 Acute angles: ∠COD, ∠COB, ∠BOA.

1 Right angle : ∠BOD.

2 Obtuse angle : ∠AOC, ∠AOD.

Question 4
Draw the letter ‘M’ such that the angles on the sides are 40° each and the angle in the middle is 60°.

Solution:

Question 5
Draw the letter ‘Y’ such that the three angles formed are 150°, 60° and 150°.

Solution:

Question 6
The Ashoka Chakra has 24 spokes. What is the degree measure of the angle between two spokes
next to each other? What is the largest acute angle formed between two spokes?

Solution:

We know that complete angle (circle) = 360

Total spokes in The Ashoka Chakra (equal parts) = 24

Therefore angle between two spokes = 360° ÷ 24 = 15°

hence, 15°  is the largest acute angle formed between two spokes.

Question 7
Puzzle: I am an acute angle. If you double my measure, you get an acute angle. If you triple my measure, you will get an acute angle again. If you quadruple (four times) my measure, you will get an acute angle yet again! But if you multiply my measure by 5, you will get an obtuse angle measure. What are the possibilities for my measure?

Solution:

Possible measure of angles =19°, 20°, 21°, 22°.