NCERT Solutions for Class 6 Maths Chapter 2 - Lines and Angles

NCERT Solutions for Class 6 Maths Chapter 2 Lines and Angles (Exercise)

Question 1:

Can you help Rihan and Sheetal find their answers?

Answer: Rihan can draw an infinite number of lines that pass through the point, whereas Sheetal can draw only one line that passes through both of the points.

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?

Answer: In the given figure, there are four line segments, i.e., $\overline{\mathrm{LM}},\overline{\mathrm{MP}},\overline{\mathrm{PQ}},\overline{\mathrm{QR}}$
Points $L$ and $R$ are on one of the line segments.
Points $M$, $P$, and $Q$ are on two of the line segments.$$

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

Answer:

In the given figure, there are two rays:

• Ray TA: This ray starts at point T and passes through point A, extending infinitely beyond A.

• Ray TB: This ray also starts at point T and passes through point B, extending infinitely beyond B.

So yes, T is the starting point of both rays.

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.

Answer:

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.

Question 5: In Fig. 2.6, name:

a. Five points

b. A line

c. Four rays

d. Five line segments

Answer:

(a) Five points are: B, C, D, E, and O

(b) A line is: DB (or BD)

(c) Four rays are: OB, OC, OD, and OE

(d) Five line segments are: OB, OC, OD, DE, and OE.

Question 6: Here is a ray $\overrightarrow{OA}$ (Fig. 2.7). It starts at O and passes through the point A. It also passes through point B.
a. Can you also name it as $\overrightarrow{OB}$? Why?
b. Can we write $\overrightarrow{OA}$ as $\overrightarrow{AO}$? Why or why not?

Answer:

a) Yes, the ray can also be named $\overrightarrow{OB}$ because the ray $\overrightarrow{OA}$ passes through point B as well. Rays are named starting from the initial point and passing through any other point on the ray. Since the ray starts at O and passes through both B and A, it can be named $\overrightarrow{OB}$.

b) No, we cannot write $\overrightarrow{OA}$ as $\overrightarrow{AO}$ because rays are directional. The ray starts at point 0 and extends through A, so $\overrightarrow{OA}$ indicates the direction from 0 to A. Writing it as $\overrightarrow{AO}$ would imply the ray starts at A and goes towards 0, which is incorrect in this context because 0 is the starting point.

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.

Answer:

The name of the vertex of ∠BDC is D.

The name of the vertex of ∠POR is O.

The name of the vertex of ∠LKM is K.

The name of the vertex of ∠AYC is Y.

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

Answer: We can draw an angle ∠RST of any measure with vertex $S$ and arms $\overrightarrow{ST}$ and $\overrightarrow{SR}$.

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

Answer: Using a single letter name like ∠P is insufficient in case of two or more angles having the same vertex. Here, ∠P can be considered for any of the three angles ∠APB,∠BPC or ∠APC. So, here it is necessary to use three letters in specifying an angle.

Question 4: Name the angles marked in the given figure.

Answer: In the given figure, two angles have been marked with a common arm $\overrightarrow{TR}$. We can name these angles as ∠RTQ and ∠RTP or ∠PTR and ∠QTR.

Question 5: Mark any three points on your paper that are not on one line. Label them A, B, and 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 $\overleftrightarrow{AB}$, $\overleftrightarrow{BC}$, and $\overleftrightarrow{CA}$ by joining pairs of these points. There are three angles, ∠ABC, ∠ACB, and ∠BAC, formed using these points.

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.

Answer: We can draw the figure as shown below. We get 6 lines in all. These are $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, $\overrightarrow{CD}$, $\overrightarrow{AC}$, $\overrightarrow{BD}$.

We get 12 angles in all. These are ∠ABC, ∠ABD, ∠GBD; ∠BCD, ∠BCA, ∠ACD; ∠CDA, ∠CDB, ∠BDA; ∠DAB, ∠DAC, and ∠CAB.

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?

Answer:

The angles formed with a line along the fold created are ∠AEF, ∠BEF,∠DFE, and ∠CFE, which are marked with 1, 2, 3, and 4. Out of these angles ∠AEF and ∠CFE are the larger ones, whereas ∠BEF and ∠DFE are the smaller ones.

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.

Answer:

(a) ∠AOB>∠XOY because ∠XOY lies in the interior of ∠AOB.

(b) ∠AOB>∠XOB because both angles have one arm, OB common, but the uncommon arm OX of ∠XOB lies in the interior of ∠AOB.

(c) ∠XOB =∠XOC because OB and OC both denote the same ray, so both angles have the common arms and the same vertex. That is, they superimpose on each other.

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

Answer:

$\begin{array}{r}\end{array}$

∠1 > ∠3 (∠1 has more spread than ∠3)

∠1 + ∠2 > ∠3 + ∠2 (adding ∠2 to both sides)

Hence, ∠XOY > ∠AOB

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

Answer: In my classroom, the windows typically contain several right angles, as the corners where the frames meet are usually 90 degrees. Every rectangular or square-shaped window has four right angles.

Besides windows, other objects in the classroom with right angles include the corners of the doors, the edges of the blackboard or whiteboard, the desks or tables, and tiles on the floor or walls may also contain right angles as they are square and rectangular.

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?

Answer: We can produce BA beyond A to join other points of the grid. Thus, we make a straight angle as shown in the figure below. Clearly, ∠CAB is a straight angle. There is only one way of doing it.

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

Hint: Extend the line further as shown in the figure below. To get a right angle at A, we need to draw a line through it that divides the straight angle CAB into two equal parts.

Answer: We can produce BA beyond A to make a straight angle, then through A, draw the line DE, which is perpendicular to BC as shown in the figure below. Clearly, ∠DAB or ∠EAB is a right angle. There is only one way of doing it.

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.

Answer: We can fold the paper as shown in the figure.

(a) We get 4 right angles. The angle formed at any point and one side of a line is always a straight angle. The perpendicular drawn by the second crease through the vertex of a straight angle bisects it into two right angles.
(b) We first fold the paper to get a slanting crease and then fold the paper again in such a way that a part of the slanting crease falls on the other part of itself. Doing it carefully, we get another slanting line which is perpendicular to the first one. Thus, we get 4 right angles at the meeting point of the two lines as shown in the figure.

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

Answer:

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

Answer: Acute angles and obtuse angles in different orientations:

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: Yes, the words acute and obtuse in geometry have meanings that relate to their everyday usage:

Acute: The word acute means "sharp" in general English, which is why an acute angle is described this way. Acute angles are smaller than 90 degrees and visually appear "sharp" or narrow, like the tip of a knife or a point, which gives a sharp impression.

Obtuse: The word obtuse means "blunt" or "dull." An obtuse angle is greater than 90 degrees and appears wider or broader, which makes it look blunt or less sharp, much like a blunt object or a broad, dull edge.

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?

Answer: We find the number of acute angles in the given figures are:

The next figure will be

So, the number pattern is 3, 3 + 9 = 12, 12 + 9 = 21, 21 + 9 = 30,…

Question 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) Yes, it is possible to count the number of units in 5s or 10s because there are also marking for each $5^{\circ }$ and ${10}^{\circ }$ on the protractor and the arms $\overrightarrow{\mathrm{AK}}$ and $\overrightarrow{\mathrm{AL}}$ pass through the ${10}^{\circ }$ markings.
(b) In the given figure, the arms $\overrightarrow{\mathrm{AL}}$ and $\overrightarrow{\mathrm{AW}}$ pass through the ${10}^{\circ }$ markings. There are five ${10}^{\circ }$ markings of scale where $\overrightarrow{\mathrm{AL}}$ and $\overrightarrow{\mathrm{AW}}$ pass. So, $\mathrm{\angle }\mathrm{WAL}={50}^{\circ }$.
(c) In the given figure, the arms $\overrightarrow{\mathrm{AK}}$ and $\overrightarrow{\mathrm{AT}}$ pass through the ${10}^{\circ }$ markings. There are twelve ${10}^{\circ }$ markings of scale where $\overrightarrow{\mathrm{AK}}$ and $\overrightarrow{\mathrm{AT}}$ pass. So, $\mathrm{\angle }\mathrm{TAK}={120}^{\circ }$.

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

Answer:

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

Answer: Take different angles and measure them.

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

Answer:

The degree measures for the angles using a paper protractor are given below:

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

Answer: We can draw a ray opposite to one of the arms and split the angle into two parts - a straight angle and an acute angle. Measure the acute angle (∠1 = 80°) using a protractor and add it to 180°. It would be the measure of the required angle (180° + 80° = 260°).

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

Answer:

(a) 80°

(b) 120°

(c) 60°

(d) 130°

(e) 125°

(f) 62°

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

Answer: The degree measures of ∠BXE, ∠CXE, ∠AXB, and ∠BXC are as follows:

∠BXE = 115°

∠CXE = 85°

∠AXB = 65°

∠BXC = 30°$\begin{array}{r}\end{array}$

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

Answer: The degree measures of ∠PQR, ∠PQS, and ∠PQT are as follows:

∠PQR=45°

∠PQS=100°

∠PQT=152°

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.

Answer:

Step 1: Fold the square diagonally to form a triangle.

Step 2: Fold the bottom tip of the triangle upwards.

Step 3: Fold the right and left corners upwards to make ears.

Step 4: Fold the side edges inward to form the sides of the face.

Step 5: Fold the ear tips outward slightly.

Step 6: Fold the top corner behind the model to round the head.

Step 7: Fold the bottom point backward.

Step 8: Draw eyes, nose, and whiskers to complete the bunny!

In the complete bunny face, we will have multiple angles:

The ears form acute angles (around 30° to 45°). Carefully unfold the entire model to return to a flat sheet.

The face near the chin forms an obtuse angle, close to 120°.

The sides of the face are around 90° to 120°, depending on the precision of the folds.

Question 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) The sum of the three angles of the triangle: 45° + 65° + 70° = 180°

b) The sum of the three angles of the triangle: 56° + 62° + 62° = 180°

c) The sum of the three angles of the triangle: 30° + 52° + 98° = 180°

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.

Answer:

a) The clock dial has been divided into 12 equal parts. And the dial of the clock represents the angle of 360°.

So, the angle measure of each part is $\frac{360°}{12}$ = 30°.

Thus, at 1 o'clock, the angle between the hands is 30°.

b) From 12 to 2, we have 2 parts at 2 o'clock. So, the angle between the hands is 60°.

Similarly, between 12 and 4, we have four parts.

So, at 4 o'clock, the angle between the hands is 120°. Similarly, at 6 o'clock, the angle between the hands is 180°.

c) The other angles made by the hands of a clock:

Angle between hands at 5o′ clock = 5 × 30° = 150°

Angle between hands at 7 o' clock = 7 × 30° = 210°

Angle between hands at 8o′ clock = 8 × 30° = 240°

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?

Answer: Yes, it is possible to express the amount by which a door is opened using an angle. The vertex of the angle will be the hinge of the door, and the arms of the angle will be the wall and the door itself.

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?

Answer: Vidya's original position is along the vertical line. When she starts swinging, the rope of the swing is along the slanting line, which makes an angle with the imaginary vertical line. The speed depends on this angle.

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?

Answer: Yes, angles can be used to describe the slopes of the slabs. Horizontal imaginary line and slab are the arms of each angle. Here, the slab arm is visible, but the horizontal line is not 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.

Answer: Observe the vertical and horizontal lines touching the insects. The rotation from vertical to horizontal position makes an angle. We can imagine the meeting point of the two lines as the vertex, and these two lines as arms of the angle, as shown in the picture.

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.

Answer:

The table to record my guesses and actual measurements:

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

Answer:

a)

b)

c)

d)

e)

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.

Answer: We can draw an angle equal to the given angle by following these steps:

Step I: Draw a ray $\overrightarrow{AB}$.

Step II: Measure the angle IHJ. It is 118°.

Step III: Place the centre of the protractor at A and the zero edge along $\overrightarrow{AB}$.

Step IV: Start counting from zero near B. Mark a point C at 118°.

Step V: Join AC.

Thus, ∠BAC = ∠IHJ.

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

b. An obtuse angle

c. A reflex angle

Mark the intended angles with curves to specify the angles. One has been done for you.

Answer:

a) Acute angle

b) Obtuse angle

c) Reflex angle

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

Answer:

Using a protractor, we find the measure of each angle and then classify each angle as follows:

(a) ∠PTR = 31°, acute angle

(b) ∠PTQ = 60°, acute angle

(c) ∠PTW = 104°, obtuse angle

(d) ∠WTP = 360° − 104° = 256°, reflex angle.

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

Answer:

a)

b)

c)

d)

e)

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.

Answer:

(a) 45°, acute angle

(b) 169°, obtuse angle

(c) 120°, obtuse angle

(d) 33°, acute angle

(e) 99°, obtuse angle

(f) 348°, reflex angle

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

Answer: Figure with three acute angles, one right angle and two obtuse angles:

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

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

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

Answer: The letter 'Y′ such that the three angles formed are 150°, 60°, and 150°.

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?

Answer: The Ashoka Chakra resembles a circle. So its degree measure is 360°.

Dividing it into 24 equal parts by spokes, the measure of each part = $\frac{360°}{24}$ = 15°

Thus, the degree measure of the angle between two spokes next to each other is 15°.

The largest acute angle formed between two spokes is 5 × 15° = 75°.

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?

Answer: As per the given condition, we have $4\times$ acute angle $<{90}^{\circ }$ and $5\times$ acute angle $>{90}^{\circ }$

$\Rightarrow$ acute angle $<\frac{{90}^{\circ }}{4}=22{\frac{1}{2}}^{\circ }$ and acute angle $>\frac{{90}^{\circ }}{5}={18}^{\circ }$

So, possible measures of the acute angle would be ${19}^{\circ },{20}^{\circ },{21}^{\circ }$ and ${22}^{\circ }$ in whole numbers.