NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 2 – Lines and Angles

NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 2 – Lines and Angles are given here in easy, step-by-step form. NCERT stands for National Council of Educational Research and Training, the body that makes your textbook. Ganita Prakash is the new Class 6 Maths book. This chapter builds the basic ideas of geometry. You will learn about a point, a line segment, a line and a ray, and then about angles – how to name them, compare them, measure them with a protractor, draw them, and sort them into types like acute, right, obtuse, straight and reflex.

The chapter is made of short activity boxes called “Figure it Out”. All the Figure it Out questions are solved below, section by section, in simple language, with clear diagrams. Students can use this page for homework and revision, parents can use it to check the work at home, and teachers can use it in class. A free printable PDF is also available for offline study.

2.1–2.4 Point, Line Segment, Line, Ray – Figure it Out (Pages 15–17)

Q.1.

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?

Can you help Rihan and Sheetal find their answers?
Answer:

• Rihan can draw many/uncountable number of lines through the given point.
• Sheetal can draw only one line through the two given points.

Q.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?

Answer: LM, MP, PQ, QR
Points L and R are exactly on one line segment. Points M, P and Q are on two line
segments.

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

Answer: TA, TB, TN and NB
No, T is the starting point of TB, TN and TA but not of NB.
Q.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.

 

Answer:

Q.5. In Fig. 2.6, name:

• a. Five points
• b. A line
• c. Four rays
• d. Five line segments

Answer:

) D, E, O, B and C

b)

$\overleftrightarrow{DE}$

or

$\overleftrightarrow{DO}$

or

$\overleftrightarrow{DB}$

or

$\overleftrightarrow{EO}$

or

$\overleftrightarrow{EB}$

or

$\overleftrightarrow{OB}$

 

c)

$\overrightarrow{OC},\ \overrightarrow{OB},\ \overrightarrow{OE},\ \overrightarrow{OD}$

(Try for other rays)

d)

$\overline{DE},\ \overline{DO},\ \overline{DB},\ \overline{EO},\ \overline{EB}$

(

$\overline{OB}$

,

$\overline{OC}$

are also possible)

Q.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?

Answer:

a) Yes, O is the starting point and point B lies on the ray that goes endlessly in the direction of A.

$\overrightarrow{OA}$

is the extension of

$\overrightarrow{OB}$

.

b) No,

$\overrightarrow{OA}$

is a ray with starting point O whereas

$\overrightarrow{AO}$

is a ray with starting point A.

2.5 Angle – Figure it Out (Pages 19–21)

Q.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.

Answer:

a):

The angle is ∠BDC. It has a vertex D. Its rays are

$\overrightarrow{DB}$

and

$\overrightarrow{DC}$

.

b):

The angle is ∠POQ. It has a vertex O. Its rays are

$\overrightarrow{OQ}$

and

$\overrightarrow{OP}$

.

c):

The angle is ∠EOF. It has a vertex O. Its rays are

$\overrightarrow{OE}$

and

$\overrightarrow{OF}$

.

d):

The angle is ∠ABC. It has a vertex B. Its rays are

$\overrightarrow{BA}$

and

$\overrightarrow{BC}$

.

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

Answer:

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

Answer:

∠APC is a combination of two angles∠APB and∠BPC. So, ∠APC cannot be labelled as ∠P because there is more than one angle at vertex P.

Q.4. Name the angles marked in the given figure.

Answer: ∠RTQ, ∠RTP

Q.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.

Answer: We get three lines

$\overleftrightarrow{AB}$

,

$\overleftrightarrow{BC}$

,

$\overleftrightarrow{CA}$

.

Using A, B & C we can name three angles: ∠ABC or ∠CBA, ∠BCA or ∠ACB & ∠CAB
or ∠BAC.

Q.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.

Answer:

We get six lines

$\overleftrightarrow{AB}$

,

$\overleftrightarrow{BC}$

,

$\overleftrightarrow{CD}$

,

$\overleftrightarrow{DA}$

,

$\overleftrightarrow{AC}$

&

$\overleftrightarrow{BD}$

.

Using A, B, C & D we can name the following angles:

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

2.6 Comparing Angles – Figure it Out (Page 23)

Q.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?

Answer:

Angles Formed: ∠AEF, ∠BEF, ∠DFE, ∠CFE
Here ∠AEF & ∠CFE are larger than ∠BEF & ∠DFE
Try more cases by folding rectangular sheets in different ways.

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

• a. ∠AOB or ∠XOY
• b. ∠AOB or ∠XOB
• c. ∠XOB or ∠XOC

Discuss with your friends on how you decided which one is greater.

Answer:

(a)- ∠AOB; ∠XOY is an acute angle and ∠AOB = ∠AOX + ∠XOY+∠YOB
(b)- ∠AOB
(c) – None. ∠XOB = ∠XOC

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

Answer:

By looking at the figure we cannot say. Superimposition or measurement is necessary here.

Q.1. Is it possible to draw 

$\overrightarrow{OC}$

 such that the two angles are equal to each other in size?

Answer:

Yes, when

$\overrightarrow{OA}$

and

$\overrightarrow{OB}$

overlap each other on folding Vidya’s notebook, the crease

$\overrightarrow{OC}$

will divide ∠AOB into two equal-sized angles.

Q.1. If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?
Answer:

$\frac{1}{4}$

of a full turn.

2.7–2.8 Rotating Arms and Special Angles – Figure it Out (Pages 29–31)

Q.1. How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?
Answer:
Each window of our classroom contains four right angles. Yes, right angles can also be seen in other objects around the classroom, such as tables, chairs, cupboards, doors, and whiteboards.

Q.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?

Answer:

Q.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?

Answer:

Q.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.

Answer:

• a. Four right angles. Each angle is
  $\frac{1}{4}$of the complete angle.
• b. Explore different ways of doing it.

2.7 Making Rotating Arms 2.8 Special Types of Angles Figure it Out (Page No. 31-32)

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

Answer:

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

Answer:

Q.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:
These words were chosen because in an acute angle, the opening of the edges is less than the obtuse angle.

Q.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?

Answer:

• 1st figure (1 triangle) → 3 acute angles
• 2nd figure (2 triangles) → 12 acute angles
• 3rd figure (3 triangles) → 21 acute angles
• So, the 4th figure (4 triangles) will have 30 acute angles.
• Since 12 – 3 = 9, 21 – 12 = 9 and 30 – 21 = 9.
• Thus, each time a new triangle is added, the number of acute angles increases by 9.

2.9 Measuring Angles Figure it Out (Page No. 35)

Q.1. Write the measures of the following angles:
a. ∠ KAL
Notice that the vertex of this angle coincides with the centre of the protractor. So the number of units of 1 degree angle between KA and AL gives the measure of ∠KAL. By counting, we get —∠KAL = 30°
Making use of the medium-sized and large-sized marks, is it possible to count the number of units in 5s or 10s?
b. ∠WAL
c. ∠TAK

Answer:

a. ∠KAL = 30°.
Yes, it is possible to count the number of units in 5s or 10s.
b. ∠WAL = 50°
c. ∠TAK = 120°

Q.2. Name the different angles in the figure and write their measures.

Answer:

∠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°,and ∠TOU = 20°.

Q.3. In Fig. 2.20, we have ∠AOB = ∠BOC = ∠COD = ∠DOE = ∠EOF = ∠FOG = ∠GOH = ∠HOI=_____. Why?

Answer:

Each of the angles = 22.5 °
Here, the straight angle of 180 ° is divided into eight equal parts.
Therefore, measurement of each angle will be

$\frac{180°}{8}$

= 22.5 °

2.9 Measuring Angles Figure it Out (Page No. 40-43)

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

Answer:
∠IHJ = ∠JHI = 47°
∠GHK = ∠IHJ = 23°
∠IHJ = ∠JHI = 108°

Q.2. Find the degree measures of different angles in your classroom using your protractor.
Answer:
Partially opened door: 45°
Corner of the window frame: 90°
Corner of the desk: 90°
Opened book lying on a desk: 180°
Backrest of a chair: 110°

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

Answer:
∠IHJ = 42°, ∠IHJ = 116°
No, a paper protractor cannot work here.

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

Answer:

Measure of unmarked angle = 100°
Measure of marked angle = 360° – Measure of unmarked angle
= 360° – 100° = 260°.

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

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

Answer:
∠BXE = 115°
∠CXE = 85°
∠AXB = 65
∠BXC = ∠AXC – ∠AXB = 95° – 65° = 30°

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

Answer:
∠PQR = 45°
∠PQS = 80°
∠PQT = 130°

Q.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.

Answer:
Do it yourself.

Q.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! We will come back to why this happens in a later year.

Answer:

(a) Sum of angles of △ABC = ∠ACB + ∠CAB + ∠A = 70° + 45° + 65° = 180°

(b) Sum of angles of △ABC = ∠ACB + ∠CAB + ∠ABC = 62° + 55° + 63°= 180

(c) Sum of angles of △ABC = ∠ACB + ∠CAB + ∠ABC = 97° + 30° + 53° = 180°
Conjecture: The sum of angles of a triangle, whether scalene, isosceles, or equilateral, is always 180°.

2.9 Measuring Angles Figure it Out (Page No. 45-46)

Q.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.

Answer:
a. The angle at the centre of the clock is 360° which is divided into 12 equal parts.
So, the angle between two successive numbers =

$\frac{360°}{12}$

= 30°.

b. At 2’O clock = 2×30 = 60°
At 4’O clock = 4×30 = 120°
At 6’O clock = 6×30 = 180°

(c) At 3 o’clock = 3 × 30° = 90°
At 5 o’clock = 5 × 30° = 150°
At 7 o’clock = 7 × 30° = 210°
At 8 o’clock = 8 × 30° = 240°
At 9 o’clock = 9 × 30° = 270°
At 10 o’clock = 10 × 30° = 300°
At 11 o’clock = 11 × 30° = 330°
At 12 o’clock = 12 × 30° = 360°

Q.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?

Answer:
Yes, the hinge of the door will be the vertex of the angle while the wall and the door will be the arms of the angle.

Q.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?

Answer:
Student may not see the angle but the angle can be thought to be between positions where she starts the swinging (the initial position) and the position where she attains the greatest position of the swing at any one side.

Q.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?

Answer:
Yes, angles can be used to describe the slopes of the slab.
The larger the angle, the greater will be the slope of the slab.
For each angle, one arm is a side, and the other arm is a slope.
The vertical arm (side) is not visible, whereas the other arm (slope) is visible.

Q.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.

Answer:
Yes, angles can be used to describe the amount of rotation by observing the initial and final positions of the insect.
The horizontal line touching the insect and the insect itself will form the arms of the angle. The vertex will be the back end of the insect.

2.10 Drawing Angles Figure it Out (Page No. 49-50)

Q.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.

Answer:
Angles are ∠PAC, ∠ACL, ∠APL, ∠CLP, ∠RPL, ∠PLS, ∠LSR, ∠PRS, ∠BRS.

The guessed measurements were quite close to the actual measurements.

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

Q.3. Draw an angle whose degree measure is the same as the angle given below:

Answer:
The measurement of the given angle is 116°, and it can be drawn using the following steps:
(i) Draw a ray

$\overrightarrow{OA}$

(ii) Place the center point of the protractor on O and align ray

$\overrightarrow{OA}$

with the zero line.
(iii) Starting from 0°, count up to 116° and mark a point B at the 116° mark.
(iv) Using a ruler, join points O and B.
∠AOB = 116° is the required angle.

2.11 Types of Angles and their Measures Figure it Out (Page No. 51-52)

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

Mark the intended angles with curves to specify the angles. One has been done for you.
Answer:
a. An acute angle

b. An obtuse angle

c. A reflex angle

Q.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
Answer:
a. ∠PTR = 30° (Acute angle)
b. ∠PTQ = 60° (Acute angle)
c. ∠PTW = 104° (Obtuse angle)
d. ∠WTP = 360° – ∠PTW = 360° – 104° = 256° (Reflex angle)

2.11 Types of Angles and their Measures Figure it Out (Page No. 53-54)

Q.1. In this figure, ∠TER = 80°. What is the measure of ∠BET? What is the measure of ∠SET?

Answer:
∠BET = 100° ∠SET = 10°

Figure it Out (Page 53 – 54)

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

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

Answer:

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

Answer:

∠A, ∠B, and ∠C are three acute angles.
∠D and ∠F are obtuse angles.
∠E is a right angle.

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

Answer:

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

Q.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?

Answer:
The Ashoka Chakra has 24 spokes and has a total of 360°.
Angle between two spokes =

$\frac{360°}{24}$

= 15°
Possible angles are 15°, 30° , 45°, 60°, 75°, 90°,105°,120°,135°,150°,165°,180°.
Thus, the largest acute angle between spokes is 75°.

Q.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?
Answer:
Let the angle be θ.
On doubling: 2θ < 90°
θ <

$\frac{90°}{2}$

θ < 45°
On tripling : 3θ < 90°
θ <

$\frac{90°}{3}$

θ < 30°
On quadrupling: 4θ < 90°
θ <

$\frac{90°}{4}$

θ < 22.5°
On multiplying by 5: 5θ > 90°
θ >

$\frac{90°}{5}$

θ > 18°
The angle θ must lie between 18° and 22.5°. So, the possible angles are: 19°, 20°, 21°, 22°.

Related Links

• Chapter 1 Patterns in Mathematics
• Chapter 3 Number Play
• Chapter 4 Data Handling and Presentation
• Chapter 5 Prime Time
• Chapter 6 Perimeter and Area
• Chapter 7 Fractions
• Chapter 8 Playing with Constructions
• Chapter 9 Symmetry
• Chapter 10 The Other Side of Zero