Class 6 Maths Chapter 2 Lines and Angles – easy notes, concepts, NCERT book solutions, important questions, worksheet and answers. Learn about lines, line segments, rays, angles and different types of angles with simple explanations.

Class 6 Mathematics Chapter 2 Lines and Angles – Complete Notes and Solutions

Chapter 2 — Lines and Angles

Class 6 Mathematics | Ganita Prakash | Complete Concepts, Diagrams and In-text Solutions

Point

Line Segment

Line

Ray

Angles

Comparing Angles

Special Angles

Protractor

Drawing Angles

Types of Angles

2.1 Point

What is a Point?

A point shows a precise location. It has no length, breadth or height.

Remember: A point is usually shown by a small dot and named by a capital letter such as P, Z or T.

Three points P, Z and T

2.2 Line Segment

Line Segment

A line segment is the shortest path joining two points.

Line segment AB has two end points A and B.

Line segment AB can be written as AB or BA.

2.3 Line

What is a Line?

A line extends endlessly in both directions.

Line AB extends endlessly in both directions.

Key idea: Two different points determine one unique line.

2.4 Ray

What is a Ray?

A ray starts at one point and goes endlessly in one direction.

Ray AP starts at A and continues endlessly through P.

Page 15 • Figure it Out • Q1

1. Can you help Rihan and Sheetal find their answers?

Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point?

Sheetal marked two points on a piece of paper. How many different lines can she draw that pass through both of the points?

Solution:

Through one point, many lines can be drawn.
Through two distinct points, exactly one line can be drawn.

Page 16 • Q2

2. Name the line segments in Fig. 2.4. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments?

Solution:

Line segments: LM, MP, PQ and QR.

Exactly one: L and R.

Exactly two: M, P and Q.

Page 16 • Q3

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

Solution:

The rays are TA, TB, TN and NB.

No. T is the starting point of TA, TB and TN. N is the starting point of NB.

Page 16 • Q4

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

a. OP and OQ meet at O.

b. XY and PQ intersect at point M.

c. Line l contains points E and F but not point D.

d. Point P lies on AB.

Solution:

The diagram above gives suitable examples.

Page 16 • Q5

5. In Fig. 2.6, name:

a. Five points

b. A line

c. Four rays

d. Five line segments

Solution:

a. Five points: D, E, O, B and C.

b. A line: DE. Other correct pairs on the same line are also possible.

c. Four rays: OC, OB, OE and OD.

d. Five line segments: DE, DO, DB, EO and EB. Other correct segments are also possible.

Page 16 • Q6

6. Here is a ray OA (Fig. 2.7). It starts at O and passes through the point A. It also passes through the point B.

a. Can you also name it as OB? Why?

b. Can we write OA as AO? Why or why not?

Solution:

a. Yes. Both OA and OB have the same starting point O and lie in the same direction.

b. No. OA starts at O, while AO starts at A.

2.5 Angle

What is an Angle?

An angle is formed by two rays having a common starting point.

∠DBE or ∠EBD. B is the vertex and BD, BE are the arms.

Vertex: Common starting point of the two rays.
Arms: The two rays forming the angle.
Important: In a three-letter name of an angle, the vertex is always the middle letter.

Page 19 • Figure it Out • Q1

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: Yes. For example, one angle can be named ∠BDC. Its vertex is D and its arms are DB and DC.

Page 20 • Q2

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

Solution: Draw two rays ST and SR with common starting point S. The angle is ∠TSR or ∠RST.

Page 20 • Q3

3. Explain why ∠APB cannot be labelled as ∠P.

Solution: At P, more than two rays are present. Writing only ∠P does not tell us which two rays form the angle. So three letters are needed.

Page 20 • Q4

4. Name the angles marked in the given figure.

Solution: ∠RTQ and ∠RTP.

Page 20 • Q5

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 as in Fig. 2.9.

Solution:

There are 3 lines:

AB, BC and CA.

There are 3 angles:

∠ABC, ∠BCA and ∠CAB.

Page 20 • Q6

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 as in Fig. 2.9.

Solution:

We get 6 lines:

AB, BC, CD, DA, AC and BD.

One complete set of named angles is:

∠BAC, ∠CAD, ∠BAD, ∠ADB, ∠BDC, ∠ADC, ∠DCA, ∠ACB, ∠DCB, ∠CBD, ∠DBA and ∠CBA.

Understanding the Size of an Angle

Angle Size Means Amount of Turn

The size of an angle depends on how much one ray has to turn to reach the other ray.

Greater turn means a larger angle.

The length of the arms does not decide the size of an angle. The amount of turn does.

Page 20 • Concept Check

Is it always easy to compare two angles?

Solution: No. Sometimes two angles look almost the same, so we need superimposition or measurement to compare them accurately.

2.6 Comparing Angles

Comparing by Superimposition

To compare two angles, place one angle over the other so that their vertices overlap.

If the arms of one angle lie exactly over the arms of another, the two angles are equal.

Page 23 • Figure it Out • Q1

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:

Do it yourself

Page 23 • Q2

2. In each case, determine which angle is greater and why.

a. ∠AOB or ∠XOY

b. ∠AOB or ∠XOB

c. ∠XOB or ∠XOC

Solution:

a. ∠AOB is greater. by Superimposition

b. ∠AOB is greater, by Superimposition.

c. ∠XOB and ∠XOC are equal, Superimposition.

Page 23 • Q3

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

Solution:

From the given figure, ∠1 is greater than ∠3 because ∠1 has a greater spread than ∠3. Adding ∠2 to both sides, we get ∠1 + ∠2 > ∠3 + ∠2. Since ∠1 + ∠2 = ∠XOY and ∠3 + ∠2 = ∠AOB, therefore, ∠XOY > ∠AOB.

2.8 Special Types of Angles

Page 28 • Concept Question

Is it possible to draw OC such that the two angles are equal to each other in size?

Solution: Yes. Fold the paper so that OA overlaps OB. The crease OC divides the straight angle into two equal angles.

Page 29 • Concept Question

If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?

Solution: A right angle is one-fourth of a full turn.

Page 29 • Figure it Out • Q1

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

Solution: Count the four corners of each rectangular window. Each corner is a right angle. Other examples include corners of books, tables, doors and boards.

Page 30 • Q2

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?

Solution:

Join A to the grid points that lie in a straight line with A and B. The exact answers depend on the shown grid.

Page 30 • Q3

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?

Solution:

Draw a line through A perpendicular to the straight line through A and B. The grid points lying on this perpendicular give the required answers.

Page 31 • Q4

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:

a. Four right angles are formed. Each is one-fourth of a complete turn.

b. do it yourself Fold the paper so that the slanting crease lies exactly over itself. Make the new crease. The two creases meet at right angles.

Page 31 • Classifying Angles • Q1

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

Solution:

do it yourself Acute angles are smaller than a right angle, right angles are exactly 90°, obtuse angles are between 90° and 180°, and straight angles are 180°.

Page 31 • Q2

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

Solution:

Page 31 • Q3

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?

Solution: Acute angles have a sharp-looking opening, while obtuse angles have a wider, more blunt-looking opening.

Page 31 • Q4

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:

The numbers are 3, 12 and 21.

The next figure will have 30 acute angles.

The pattern increases by 9 each time.

2.9 Measuring Angles

Degree Measure

A complete turn is divided into 360 equal parts. Each part is 1 degree.

Full turn = 360°
Straight angle = 180°
Right angle = 90°

Page 35 • Figure it Out • Q1

1. Write the measures of the following angles:

a. ∠KAL

b. ∠WAL

c. ∠TAK

Solution:

a. ∠KAL = 30°

b. ∠WAL = 50°

c. ∠TAK = 120°

Page 36 • Figure it Out

Name the different angles in the figure and write their measures.

Solution:

Angle	Measure	Angle	Measure
∠POQ	35°	∠POR	95°
∠POS	125°	∠POT	160°
∠QOR	60°	∠QOS	90°
∠QOT	125°	∠QOU	145°
∠ROS	30°	∠ROT	65°
∠ROU	85°	∠SOT	35°
∠SOU	55°	∠TOU	20°

Page 37 • Concept Check

What is the degree measure of ∠AOB?

Solution: Place the centre of the protractor at O and align one arm with 0°. Read the scale where the other arm crosses. That reading is the measure of ∠AOB.

Page 38 • Make Your Own Protractor

Complete the missing angle measures.

Solution:

Half turn = 180°.

Quarter turn = 90°.

Therefore, the blanks are 180° and 90°.

Page 39 • Step 5

Continuing with another half fold as shown in Fig. 2.18, we get an angle of measure ______.

Solution: 22.5°.

Page 40 • Think!

In Fig. 2.20, we have ∠AOB = ∠BOC = ∠COD = ∠DOE = ∠EOF = ∠FOG = ∠GOH = ∠HOI = _____ . Why?

Solution:

Each angle is 22.5°.

The straight angle of 180° is divided into 8 equal parts:

180° ÷ 8 = 22.5°

Page 40 • Figure it Out • Q1

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

Solution:

The shown angles are approximately 47°, 23° and 108°, according to their respective diagrams.

Page 41 • Q3

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

Solution:

The given angles are 42° and 116°.

A paper protractor cannot be used for the angle that extends beyond the semicircular measuring range in the shown figure.

Page 41 • Q4

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

Solution:

Measure the smaller unmarked angle first.

Marked reflex angle = 360° − smaller angle

For the shown example:

360° − 100° = 260°

Page 41–42 • Q5

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

Solution:

Part	Measure
a	80°
b	120°
c	60°
d	130°
e	130°
f	60°

Page 42 • Q6

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

Solution:

∠BXE = 115°

∠CXE = 85°

∠AXB = 65°

∠BXC = 30°

Page 42 • Q7

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

Solution:

∠PQR = 45°

∠PQS = 100°

∠PQT = 150°

Page 43 • Q8

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: Follow the folding sequence carefully. After opening the paper, draw over every crease and use a protractor to measure each angle.

Page 43 • Q9

9. Measure all three angles of the triangle shown in Fig. 2.21 (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 Fig. 2.21 (b) and (c). Try it for other triangles as well, and then make a conjecture for what happens in general!

Solution: Measure the three angles and add them. In each triangle, the total is 180°. This suggests the conjecture that the three angles of a triangle add up to 180°.

Mind the Mistake, Mend the Mistake!

Page 44

A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected.

Solution:

The centre of the protractor must be exactly on the vertex.
One arm of the angle must be aligned with the 0° line.
Always read the scale that begins with 0° on the arm being used.
For a reflex angle, measure the smaller angle and subtract it from 360°.

Where Are the Angles?

Page 45 • Q1

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 o’clock? And at 4 o’clock? 6 o’clock?

c. Explore other angles made by the hands of a clock.

Solution:

a. A full circle is 360° and a clock has 12 equal divisions.

360° ÷ 12 = 30°

So, each hour mark is 30° apart.

2 o’clock = 60°
4 o’clock = 120°
6 o’clock = 180°

c. For example, 3 o’clock = 90° and 9 o’clock = 270° for the reflex angle.

Page 45 • Q2

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. The vertex is the point where the door is hinged. The arms are along the door and the fixed wall or reference position.

Page 45 • Q3

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: The angle can be thought of between the initial position of the swing and its displaced position. The top support acts as the point around which the swing turns.

Page 46 • Q4

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. The slope can be described by an angle. One arm is the slab and the other is the horizontal reference line. The horizontal reference arm may not be visible.

Page 46 • Q5

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?

Solution: Yes. The angle between the original direction and the rotated direction represents the amount of rotation. The arms are the original and new directions, and the vertex is the common reference point.

2.10 Drawing Angles

Page 49 • Figure it Out • Q1

1. In Fig. 2.23, 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:

Some of the possible named angles are:

∠CAP, ∠ACD, ∠APL, ∠DLP, ∠RPL, ∠SLP, ∠PRS, ∠LSR, ∠BRS and ∠CLP.

Students should first estimate and then check each angle using a protractor.

Page 50 • Q2

2. Use a protractor to draw angles having the following degree measures:

a. 110°

b. 40°

c. 75°

d. 112°

e. 134°

Solution:

Draw a base ray, place the protractor centre on the vertex, align the base with 0°, mark the required degree and join the vertex to the mark.

Page 50 • Q3

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:

Place the protractor at the vertex of the given angle.
Read its degree measure.
Draw a new base ray.
Place the protractor correctly.
Mark the same degree measure.
Join the vertex to the mark.

2.11 Types of Angles and Their Measures

Page 51-52 • figure it out

Q1. in each of the below grid , join A to other grid points in the figure by a straight line to get

Solution:

Page 52 • Q2
2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right or reflex.

Solution:
Angle Measure Type
∠PTR 30° Acute
∠PTQ 60° Acute
∠PTW 102° Obtuse
∠WTP 258° Reflex

Angle	Measure	Type
∠PTR	30°	Acute
∠PTQ	60°	Acute
∠PTW	102°	Obtuse
∠WTP	258°	Reflex

Page 53 • Let’s Explore
In this figure, ∠TER = 80°. What is the measure of ∠BET? What is the measure of ∠SET?

Solution:
Since ∠REB is a straight angle:
∠BET = 180° − 80° = 100°
Since ∠BES is a right angle:
∠SET = 90° − 80° = 10°

Concept	Easy Meaning
Point	Shows a precise location.
Line Segment	Shortest path joining two points.
Line	Extends endlessly in both directions.
Ray	Starts at one point and extends endlessly in one direction.
Angle	Two rays with a common starting point.
Vertex	Common starting point of an angle.
Acute Angle	Greater than 0° and less than 90°.
Right Angle	Exactly 90°.
Obtuse Angle	Greater than 90° and less than 180°.
Straight Angle	Exactly 180°.
Reflex Angle	Greater than 180° and less than 360°.
Full Turn	360°.
Protractor	Tool used to measure and draw angles.

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

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

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

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?

⭐ Remember These Points

⭐ Important Angle Facts

⭐ Easy Memory Trick

Acute = Small

Right = Exactly 90°

Obtuse = Open Wide

Straight = Straight Line

Reflex = More Than Straight

Full Turn

Chapter 2 Notes & Book Solutions

Class 6 Maths Chapter 2 – Lines and Angles | Notes, Solutions & Worksheet

Chapter 2 — Lines and Angles

What is a Point?

Line Segment

What is a Line?

What is a Ray?

1. Can you help Rihan and Sheetal find their answers?

2. Name the line segments in Fig. 2.4. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments?

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

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

5. In Fig. 2.6, name:

6. Here is a ray OA (Fig. 2.7). It starts at O and passes through the point A. It also passes through the point B.

What is an Angle?

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.

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

3. Explain why ∠APB cannot be labelled as ∠P.

4. Name the angles marked in the given figure.

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 as in Fig. 2.9.

Angle Size Means Amount of Turn

Is it always easy to compare two angles?

Comparing by Superimposition

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?

2. In each case, determine which angle is greater and why.

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

Is it possible to draw OC such that the two angles are equal to each other in size?

If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?

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

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?

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?

4. Get a slanting crease on the paper. Now, try to get another crease that is perpendicular to the slanting crease.

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

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

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?

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?

Degree Measure

1. Write the measures of the following angles:

Name the different angles in the figure and write their measures.

What is the degree measure of ∠AOB?

Complete the missing angle measures.

Continuing with another half fold as shown in Fig. 2.18, we get an angle of measure ______.

In Fig. 2.20, we have ∠AOB = ∠BOC = ∠COD = ∠DOE = ∠EOF = ∠FOG = ∠GOH = ∠HOI = _____ . Why?

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

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

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

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

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

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

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.

A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected.

1. Angles in a clock:

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?

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?

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?

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?

1. In Fig. 2.23, 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.

2. Use a protractor to draw angles having the following degree measures:

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.

Q1. in each of the below grid , join A to other grid points in the figure by a straight line to get

2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right or reflex.

In this figure, ∠TER = 80°. What is the measure of ∠BET? What is the measure of ∠SET?

1. Draw angles with the following degree measures:

2. Estimate the size of each angle and then measure it with a protractor. Classify these angles as acute, right, obtuse or reflex angles.

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

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

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

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?

⭐ Remember These Points

⭐ Important Angle Facts

⭐ Easy Memory Trick

Acute = Small

Right = Exactly 90°

Obtuse = Open Wide

Straight = Straight Line

Reflex = More Than Straight

Full Turn

Chapter 2 Notes & Book Solutions