Class 6 Maths Chapter 2 Solutions: Lines and Angles

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Solution sheet 1

Page No. 15 · Figure it Out

NCERT Class 6 Maths Chapter 2, question 1 — Source question

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?

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

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NCERT Class 6 Maths Chapter 2, question 2 — Source question

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?

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

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NCERT Class 6 Maths Chapter 2, question 3 — Source question

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

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

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NCERT Class 6 Maths Chapter 2, question 4 — Source question

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.

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[1]

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Solution sheet 2

Source solution page 44

NCERT Class 6 Maths Chapter 2, question 5 — Source question

5. In Fig. 2.6, name: a. Five points b. A line c. Four rays d. Five line segments

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a) D, E, O, B and C b) DE or DO or DB or EO or EB or OB c) OC, OB, OE, OD (Try for other rays) d) DE, DO, DB, EO, EB (OB; OC are also possible)

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NCERT Class 6 Maths Chapter 2, question 6 — Source question

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?

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a) Yes, O is the starting point and point B lies on the rays that goes endlessly in the direction of A. OA is the extension of OB. b) No, OA is a ray with starting point O whereas AO is a ray with starting point A. [2] Section 2.5 Page – 19 Figure it Out

Source worked solution for Class 6 Chapter 2, question 6, part 1 — Source worked answer · part 1

Source worked solution for Class 6 Chapter 2, question 6, part 2 — Source worked answer · part 2

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Solution sheet 3

Page – 19 · Figure it Out

NCERT Class 6 Maths Chapter 2, question 7 — Source 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.

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Yes, one of the angles is ∠BDC. It’s vertex is D. One ray is DC and the other rays DB. Try for other pictures.

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NCERT Class 6 Maths Chapter 2, question 8 — Source question

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

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Refer to the worked source solution below.

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Page – 19 · Figure it Out

NCERT Class 6 Maths Chapter 2, question 9 — Source question

4. Name the angles marked in the given figure.

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∠RTQ, ∠RTP

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NCERT Class 6 Maths Chapter 2, question 10 — Source 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 as in Fig. 2.9.

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We get three lines AB, BC, CA. Using A, B & C we can name three angles: ∠ABC or ∠CBA, ∠BCA or ∠ACB & ∠CAB or ∠BAC. [3]

Source worked solution for Class 6 Chapter 2, question 10, part 1 — Source worked answer · part 1

Source worked solution for Class 6 Chapter 2, question 10, part 2 — Source worked answer · part 2

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Solution sheet 4

Source solution page 46

NCERT Class 6 Maths Chapter 2, question 11 — Source 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 as in Fig. 2.9.

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We get six lines AB, BC, CD, DA, AC & BD Using A, B, C & D we can name the following angles ∠BAC, ∠CAD, ∠BAD, ∠ADB, ∠BDC, ∠ADC, ∠DCA, ∠ACB, ∠DCB, ∠CBD, ∠DBA & ∠CBA Section 2.6 Page 20 Is it always easy to compare two angles? No. it is not always easy to compare two angles. For eg. 89º & 91º angles cannot be compared without measuring or overlapping. But for the given figures, comparison is easy. Page 23 Where else do we use superimposition to compare? A few examples are- line segments, squares and circles. Think of more. Figure it out

Source worked solution for Class 6 Chapter 2, question 11, part 1 — Source worked answer · part 1

A few examples are- line segments, squares and circles. Think of more. · Figure it out

NCERT Class 6 Maths Chapter 2, question 12 — Source 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? [4]

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

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Source worked solution for Class 6 Chapter 2, question 12, part 2 — Source worked answer · part 2

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Solution sheet 5

Source solution page 47

NCERT Class 6 Maths Chapter 2, question 13 — Source question

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.

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

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NCERT Class 6 Maths Chapter 2, question 14 — Source question

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

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By looking at the figure we cannot say. Superimposition or measurement is necessary here. Section 2.8 Page 28

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Section 2.8 · Page 28

NCERT Class 6 Maths Chapter 2, question 15 — Source question

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

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Yes, when OA and OB overlap each other on folding the Vidya’s notebook, the crease OC will divide ∠AOB in two equal sized angles. Page 29

Source worked solution for Class 6 Chapter 2, question 15, part 1 — Source worked answer · part 1

Page 28 · Page 29

NCERT Class 6 Maths Chapter 2, question 16 — Source question

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

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1 4 of a full turn. [5] Section 2.8 Page No. 29 Figure it Out

Source worked solution for Class 6 Chapter 2, question 16, part 1 — Source worked answer · part 1

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Solution sheet 6

Page No. 29 · Figure it Out

NCERT Class 6 Maths Chapter 2, question 17 — Source 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.

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a. Four right angle. Each angle is ¼ the of the complete angle. b. Explore different ways of doing it. Page 31 Figure it Out

Source worked solution for Class 6 Chapter 2, question 17, part 1 — Source worked answer · part 1

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NCERT Class 6 Maths Chapter 2, question 18 — Source question

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

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Acute angle Obtuse angle

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Page 31 · Figure it Out

NCERT Class 6 Maths Chapter 2, question 19 — Source 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?

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In acute angles the opening of the edges is lesser than the obtuse angle which have larger openings.

Source worked solution for Class 6 Chapter 2, question 19, part 1 — Source worked answer · part 1

means blunt. Why do you think these words have been chosen?

NCERT Class 6 Maths Chapter 2, question 20 — Source question

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?

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(i) three (ii) Twelve (iii) Twenty-one The next figure will have thirty acute angles. Yes, the pattern is 3 × 0 + 1, 3 × 1 + 1, 3 × 2 + 1, 3 × 3 + 1,… The number 0,1,2,3,4,….. are number of inner triangles. [6] Section 2.9 Page No. 35 Figure it out

Source worked solution for Class 6 Chapter 2, question 20, part 1 — Source worked answer · part 1

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Solution sheet 7

Page No. 35 · Figure it out

NCERT Class 6 Maths Chapter 2, question 21 — Source question

1. Write the measures of the following angles: a. ∠ KAL b. ∠WAL c. ∠TAK

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a. ∠KAL = 30° Yes, it is possible to count the number of units in 5s or 10s. b. ∠WAL = 50° c. ∠TAK = 120° Page No. 36

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Page No. 36

NCERT Class 6 Maths Chapter 2, question 22 — Source question

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

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∠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° [7] Page No. 40 Think!

Source worked solution for Class 6 Chapter 2, question 22, part 1 — Source worked answer · part 1

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Solution sheet 8

Page No. 40 · Think!

NCERT Class 6 Maths Chapter 2, question 23 — Source question

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

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Each of the angles = 22.5º As the straight angle of 180º is divided into eight equal parts so each of the right angles 180º will be of measure = 8 = 22.5º Figure it Out

Source worked solution for Class 6 Chapter 2, question 23, part 1 — Source worked answer · part 1

Figure it Out

NCERT Class 6 Maths Chapter 2, question 24 — Source question

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

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∠IHJ = ∠JHI = 47º ∠GHK = ∠IHJ = 23º ∠IHJ = ∠JHI = 108º

Source worked solution for Class 6 Chapter 2, question 24, part 1 — Source worked answer · part 1

Figure it Out

NCERT Class 6 Maths Chapter 2, question 25 — Source question

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

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∠IHJ = 42º, ∠IHJ = 116º No, paper protractor cannot work here.

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NCERT Class 6 Maths Chapter 2, question 26 — Source question

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

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Measure of marked angle = 360º – Measure of unmarked angle = 360º - 100º = 260º Try other ways to find the marked angle

Source worked solution for Class 6 Chapter 2, question 26, part 1 — Source worked answer · part 1

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NCERT Class 6 Maths Chapter 2, question 27 — Source question

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

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a. 80º b. 120º c. 60º d. 130º e. 130º f. 60º

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NCERT Class 6 Maths Chapter 2, question 28 — Source question

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

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∠BXE = 115º, ∠CXE = 85º, ∠AXB = 65º ∠BXC = 30º

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NCERT Class 6 Maths Chapter 2, question 29 — Source question

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

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∠PQR =45º ∠PQS=100º ∠PQT=150º. [8] Page 45 Figure it Out

Source worked solution for Class 6 Chapter 2, question 29, part 1 — Source worked answer · part 1

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Solution sheet 9

Page 45 · Figure it Out

NCERT Class 6 Maths Chapter 2, question 30 — Source 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 o’clock? And at 4 o’clock? 6 o’clock? c. Explore other angles made by the hands of a clock.

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(a) The angles at the centre of the clock is 360º which is divided into 12 equal parts. So. 360 The angle between two successive numbers = 12 = 30º (b) At 2’O clock = 60º = 2×30 At 4’O clock = 120º = 4×30 At 6’O clock = 180º = 6×30 (c) At 3’O clock = 90º At 9’O clock = 270º Try for other angles made by the hands of a clock.

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NCERT Class 6 Maths Chapter 2, question 31 — Source 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?

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Yes, the vertex of the angle will be the point where door meets the wall. The arms will be the edges of the door and the wall.

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NCERT Class 6 Maths Chapter 2, question 32 — Source 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?

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Student may not see the angle, but when the starting arm is fixed as the position where she starts swinging. 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.

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NCERT Class 6 Maths Chapter 2, question 33 — Source 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?

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Yes, angles can be used directly to describe slopes of the slab, larger the angle, greater the slope of the slab. For each angle, one arm is a side and one arm is the slope. The vertical arm is not visible, whereas the other arm is visible. Here in this toy, edges of the slabs are the arms of the angles. Top horizontal ray is not visible, other arms in the form of edges of the slab are visible. Teacher should motivate students to get other possible answers. [9] Page 49 Section 2.10 Figure it Out

Source worked solution for Class 6 Chapter 2, question 33, part 1 — Source worked answer · part 1

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Solution sheet 10

Section 2.10 · Figure it Out

NCERT Class 6 Maths Chapter 2, question 34 — Source question

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.

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∠CAP, ∠ACD, ∠APL, ∠DLP, ∠RPL, ∠SLP, ∠PRS, ∠LSR, ∠BRS, ∠CLP Try more! Page 52 Section 2.11 Figure it Out

Source worked solution for Class 6 Chapter 2, question 34, part 1 — Source worked answer · part 1

Section 2.11 · Figure it Out

NCERT Class 6 Maths Chapter 2, question 35 — Source question

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

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a. ∠PTR = 30º (Acute angle) b. ∠PTQ = 60º (Acute angle) c. ∠PTW = 102º (Obtuse angle) d. ∠WTP = 258º (Reflex angle) Let’s Explore:

Source worked solution for Class 6 Chapter 2, question 35, part 1 — Source worked answer · part 1

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NCERT Class 6 Maths Chapter 2, question 36 — Source question

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

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∠BET = 100º, ∠SET = 10º Page – 53 Figure it out

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NCERT Class 6 Maths Chapter 2, question 37 — Source question

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

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∠A, ∠B & ∠C are three acute angles ∠D, ∠F are obtuse angle ∠E is right angle [10]

Source worked solution for Class 6 Chapter 2, question 37, part 1 — Source worked answer · part 1

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Solution sheet 11

Page – 53 · Figure it out

NCERT Class 6 Maths Chapter 2, question 38 — Source question

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

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Refer to the worked source solution below.

Source worked solution for Class 6 Chapter 2, question 38, part 1 — Source worked answer · part 1

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NCERT Class 6 Maths Chapter 2, question 39 — Source question

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

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Refer to the worked source solution below.

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NCERT Class 6 Maths Chapter 2, question 40 — Source 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?

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The angle between two spokes next to each other is 15º. Largest acute angle between spokes is 75º.

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NCERT Class 6 Maths Chapter 2, question 41 — Source 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?

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The acute angle can be 19º, 20º, 21º & 22º. [11]

Source worked solution for Class 6 Chapter 2, question 41, part 1 — Source worked answer · part 1

Match, attempt, then check

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?

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: 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.

5. In Fig. 2.6, name: a. Five points b. A line c. Four rays d. Five line segments

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?

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.

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.

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? [4]

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.

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?

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?

1. Write the measures of the following angles: a. ∠ KAL b. ∠WAL c. ∠TAK

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

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.

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.

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?

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 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?

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?