Chapter 2

NCERT Solutions - Class 6 Mathematics

Chapter 2: Lines and Angles

Page No. 15

Section 2.4 — Figure it Out

Q1. 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?

Ans. Rihan can draw many / uncountable number of lines through the given point[cite: 2]. Sheetal can draw only one line through the two given points[cite: 2].

Q2. 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?

Ans.

• Line segments: LM, MP, PQ, QR[cite: 2]
• Points L and R are exactly on one line segment[cite: 2].
• Points M, P, and Q are on two line segments[cite: 2].

Q3. Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays?

Ans. Rays shown: TA, TB, TN, and NB[cite: 2].
No, T is the starting point of TB, TN, and TA, but not of NB[cite: 2].

Q4. 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.

Ans. Refer to figures (a), (b), (c), and (d) as provided in the textbook[cite: 2].

Q5. In Fig. 2.6, name:
a. Five points
b. A line
c. Four rays
d. Five line segments

Ans.

• a. Five points: D, E, O, B, and C[cite: 2]
• b. A line: DE, DO, DB, EO, EB, or OB[cite: 2]
• c. Four rays: OB, OE, OD, etc.[cite: 2]
• d. Five line segments: DE, DO, DB, EO, EB (OC and OB are also possible)[cite: 2]

Q6. 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?

Ans.
a. Yes, O is the starting point and point B lies on the ray that goes endlessly in the direction of A[cite: 2]. OA is the extension of OB[cite: 2].
b. No, OA is a ray with starting point O, whereas AO is a ray with starting point A[cite: 2].

Page No. 19

Section 2.5 — Figure it Out

Q1. 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.

Ans. Yes, one of the angles is ∠BDC[cite: 2]. Its vertex is D[cite: 2]. One ray is DC and the other ray is DB[cite: 2].

Q2. Draw and label an angle with arms ST and SR.

Ans. Draw an angle with vertex S and rays extending to points T and R[cite: 2].

Q4. Name the angles marked in the given figure.

Ans. ∠RTO and ∠RTP[cite: 2].

Q5. 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.

Ans.

• We get three lines: AB, BC, and CA[cite: 2].
• Using A, B & C we can name three angles: ∠ABC (or ∠CBA), ∠BCA (or ∠ACB), and ∠CAB (or ∠BAC)[cite: 2].

Q6. 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.

Ans.

• We get six lines: AB, BC, CD, DA, AC, and BD[cite: 2].
• Angles formed: ∠BAC, ∠CAD, ∠BAD, ∠ADB, ∠BDC, ∠ADC, ∠DCA, ∠ACB, ∠DCB, ∠CBD, ∠DBA, and ∠CBA[cite: 2].

Page No. 20 & 23

Section 2.6 — Figure it Out

Q. Is it always easy to compare two angles?

Ans. No, it is not always easy to compare two angles[cite: 2]. For example, 89° and 91° angles cannot be compared without measuring or overlapping[cite: 2].

Q. Where else do we use superimposition to compare?

Ans. A few examples are line segments, squares, and circles[cite: 2].

Q1. 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?

Ans. Angles Formed: ∠AEF, ∠BEF, ∠DFE, and ∠CFE[cite: 2].
Here, ∠AEF and ∠CFE are larger than ∠BEF and ∠DFE[cite: 2].

Q2. In each case, determine which angle is greater and why:
a. ∠AOB or ∠XOY
b. ∠AOB or ∠XOB
c. ∠XOB or ∠XOC

Ans.

• (a) ∠AOB is greater; ∠XOY is an acute angle and ∠AOB = ∠AOX + ∠XOY + ∠YOB[cite: 2].
• (b) ∠AOB is greater[cite: 2].
• (c) Neither; ∠XOB = ∠XOC[cite: 2].

Q3. Which angle is greater: ∠XOY or ∠AOB? Give reasons.

Ans. By looking at the figure we cannot say[cite: 2]. Superimposition or measurement is necessary here[cite: 2].

Page No. 28 - 31

Section 2.8 — Figure it Out

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

Ans. Yes, when OA and OB overlap each other on folding Vidya's notebook, the crease OC will divide ∠AOB into two equal-sized angles[cite: 2].

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

Ans. 1/4 of a full turn[cite: 2].

Q4. 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.

Ans.
a. Four right angles[cite: 2]. Each angle is 1/4 of the complete angle[cite: 2].
b. Explore different ways of doing it[cite: 2].

Q3. 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?

Ans. In acute angles, the opening of the edges is smaller than in obtuse angles, which have larger openings[cite: 2].

Q4. 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?

Ans.

• (i) Three[cite: 2]
• (ii) Twelve[cite: 2]
• (iii) Twenty-one[cite: 2]

The next figure will have 30 acute angles[cite: 2].
Pattern: 3×0+1, 3×1+1, 3×2+1, 3×3+1, ... where numbers 0, 1, 2, 3, 4 represent the number of inner triangles[cite: 2].

Page No. 35 - 45

Section 2.9 — Figure it Out

Q1. Write the measures of the following angles:
a. ∠KAL
b. ∠WAL
c. ∠TAK

Ans.

• a. ∠KAL = 30°[cite: 2] (It is possible to count units in 5s or 10s)[cite: 2].
• b. ∠WAL = 50°[cite: 2]
• c. ∠TAK = 120°[cite: 2]

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

Ans. ∠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°[cite: 2]

Think! Why are the angles equal in Fig. 2.20?

Ans. Each angle measure = 22.5°[cite: 2]. As the straight angle of 180° is divided into eight equal parts, each part measures 180° / 8 = 22.5°[cite: 2].

Q4. How can you find the degree measure of the angle given below using a protractor?

Ans. Measure of marked angle = 360° - (Measure of unmarked angle) = 360° - 100° = 260°[cite: 2].

Q1. Angles in a clock:
a. Why is the angle between the hands at 1 o'clock equal to 30°?
b. What will be the angle at 2 o'clock? 4 o'clock? 6 o'clock?
c. Explore other angles.

Ans.
a. The total angle at the centre is 360°, divided into 12 equal parts[cite: 2]. Angle between two numbers = 360° / 12 = 30°[cite: 2].
b. At 2 o'clock = 60° (2 × 30°); At 4 o'clock = 120° (4 × 30°); At 6 o'clock = 180° (6 × 30°)[cite: 2].
c. At 3 o'clock = 90°; At 9 o'clock = 270°[cite: 2].

Q2. 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 and arms?

Ans. Yes, the vertex is the point where the door meets the wall[cite: 2]. The arms are the edges of the door and the wall[cite: 2].

Page No. 52 - 53

Section 2.11 — Figure it Out

Q2. Classify each angle as acute, obtuse, right or reflex:
a. ∠PTR = 30°
b. ∠PTQ = 60°
c. ∠PTW = 102°
d. ∠WTP = 258°

Ans.

• a. Acute angle[cite: 2]
• b. Acute angle[cite: 2]
• c. Obtuse angle[cite: 2]
• d. Reflex angle[cite: 2]

Q4. Draw the letter 'M' such that the angles on the sides are 40° each and the angle in the middle is 60°.

Ans. Side angles = 40° each; Middle angle = 60°[cite: 2].

Q5. Draw the letter 'Y' such that the three angles formed are 150°, 60°, and 150°.

Ans. Top angle = 60°; Lower side angles = 150° each[cite: 2].

Q6. 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?

Ans.

• Angle between adjacent spokes = 360° / 24 = 15°[cite: 2].
• Largest acute angle between spokes = 75°[cite: 2].

Q7. Puzzle: I am an acute angle. If you double my measure, you get an acute angle. If you triple my measure, you get an acute angle again. If you quadruple my measure, you get an acute angle yet again! But if you multiply my measure by 5, you get an obtuse angle. What are the possibilities for my measure?

Ans. The possibilities for the acute angle measure are 19°, 20°, 21°, or 22°[cite: 2].