Lines and Angles NCERT Solutions | Mathematics (Maths) Class 6 PDF Download

Page 15

Q1: Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point?
Ans: Rihan can draw an infinite number of lines that pass through a single point.

Explanation: When you have just one point on a paper, you can draw as many lines as you want that pass through this point because there is no limit to how many directions a line can go from that point. Imagine a spinning top; as it spins, it covers every possible direction from that point.

Q2: Sheetal marked two points on a piece of paper. How many different lines can she draw that pass through both of the points?
Ans: Sheetal can draw exactly one line that passes through both points.

Explanation: When you have two distinct points on a paper, only one straight line can connect them directly. Think of it like a straight path between two places; there is only one shortest path that directly connects them without any curves.

Page 16

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:

• The line segments are .
• The points on exactly one line segment are L and R. 
• The points on two line segments are M, P, and Q.

Explanation: Point L is only on segment LM, and point R is only on segment QR because they are at the ends of these segments and do not connect to any other segments.
Point M connects LM and MP, point P connects MP and PQ, and point Q connects PQ and QR. These points are where two segments meet or overlap.

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

Ans:

• The rays are .
• Yes, T is the starting point of both rays. 

Explanation: A ray starts at one point and goes off infinitely in one direction. In this case, rays  start at point T and go towards A, N and B, respectively.

Q4: Draw a rough figure and write labels appropriately to illustrate each of the following:
a.  meet at O. 
Ans: 

Explanation: Draw two lines (rays) starting from point O, one going towards point P and the other towards point Q. They meet at O because it is their common starting point.

b.  intersect at point M. 
Ans:

Explanation: Draw two lines that cross each other at point M. One line goes from X to Y, and the other goes from P to Q. The point where they cross is M.

c. Line I contains points E and F but not point D. 
Ans: 

Explanation: Draw a straight line labeled l with two points, E and F, on it. Point D should be somewhere off this line.

d. Point P lies on AB.
Ans: 

Explanation: Draw a straight line segment AB with a point P located somewhere on the line between A and B.

Q5: In Fig. 2.6, name:

a. Five points
Ans: The five points in the figure are D, E, O, C, and B. 

Explanation: Points are specific locations on the diagram, and they are labeled with letters.

b. A line
Ans: The line in the figure is . 

Explanation: A line is straight and extends in both directions without ending. In the diagram, DB extends through points D and B.

c. Four rays
Ans: The four rays in the figure are . 

Explanation: A ray starts at one point and extends infinitely in one direction. Here, all rays start at point O and go through the points D, E, C, and B respectively.

d. Five line segments
Ans: The five line segments in the figure are . 

Explanation: A line segment has two endpoints, so it is like a piece of a line that doesn’t go on forever. Each of these segments connects two of the points mentioned.

Page 17

Q6: Here is a ray  (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 ? Why? 
Ans: No, you cannot name the ray as . 

Explanation: A ray is named starting from its endpoint and going through another point in its path. Since   starts at O and passes through A, it must be named , even if it passes through other points like B later on.  would imply that the ray starts at O and goes through B first.

b. Can we write ? Why or why not? 
Ans: No, we cannot write . 

Explanation: The order of letters in a ray's name matters because the first letter (O) indicates where the ray begins. Writing it as   would incorrectly suggest that A is the starting point.

Page 19

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: In ∠BDC, Rays are Vertex is D.
Ans: In ∠POQ, Rays are  Vertex is O.
Ans: In ∠EOF, Rays are  Vertex is O.
Ans: In ∠ABC, Rays are  Vertex is B

Explanation: For example, if you see two lines or edges meeting at a point, the point where they meet is the vertex of the angle. Draw two rays starting from this vertex along the lines or edges to form the angle.

Page 20

Q2: Draw and label an angle with arms ST and SR.
Ans: 
Steps:

• First, draw a point S, which will be the vertex.
• Next, draw a ray starting at S and going in one direction. Label the other end T (making the ray ST).
• Then, draw another ray starting at S but going in a different direction. Label the other end R (making the ray SR).
• This forms an angle ∠TSR with S as the vertex, ST and SR as the arms.

Explanation: The arms are the rays extending from the vertex, and the vertex is where these rays meet.

Q3: Explain why ∠APC cannot be labelled as ∠P.

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

Explanation: If you only write ∠P, it’s unclear which angle you are talking about because there could be multiple angles that share the same vertex (P). By using all three points (A, P, C), you make it clear which angle is being discussed.

Q4: Name the angles marked in the given figure.

Ans: The angles marked in the figure are:

• ∠RTQ
• ∠RTP

Explanation: To name an angle, you use three points: one on each arm and the vertex in the middle. The vertex is where the two rays or arms meet.

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 as in Fig. 2.9.
Ans: 
Steps:

• Draw three points A, B, and C on your paper.
• Draw lines connecting each pair: .

You get three lines: AB, BC, and AC.

Explanation: Each pair of points forms one line. Since there are three points, there are three lines.

• The angles formed are .

Explanation: With three points, three angles are formed at the intersections of these lines. The vertex of each angle is the point where the two rays meet.

Page 21

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, and mark each of them with a curve as in Fig. 2.9.
Ans: 
Steps:

• Draw four points A, B, C, and D on your paper.
• Draw lines connecting each pair: .

You get six lines: .

Explanation: With four points, each pair of points forms one line, resulting in six lines.

• 12 angles are formed which are ∠DAB, ∠ABC, ∠BCD, ∠CDA, ∠CAB, ∠ABD, ∠DBC, ∠BCA, ∠ACD, ∠CDB, ∠BDA, ∠DAC.

Explanation: With four points, several angles can be formed at the intersections of these lines. Each angle is named by three points, with the vertex in the middle.

Page 23

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:

• All angles are ∠DAB, ∠ABC, ∠BCD, ∠CDA, ∠AEF, ∠BFE, ∠DEF, ∠CFE. 
• On comparing,
  ∠AEF ≥ ∠DEF
  ∠BFE ≤ ∠CFE.

Explanation: By folding the paper, you create an angle at the fold. The largest angle you can make by folding is a straight angle (180°) when the fold lies flat along one side of the paper. The smallest angle is when the fold barely moves away from the side, creating a very acute angle.

Q2: 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.
Ans:
a. ∠AOB > ∠XOY because ∠AOB opens wider than ∠XOY.
b. ∠AOB > ∠XOB because ∠AOB opens wider than ∠XOB.
c. ∠XOB = ∠XOC because arms of both angles are equal distance apart.

Explanation: a. When comparing two angles, the one that opens wider is larger.
b. ∠AOB covers more space between its arms than ∠XOB.
c. You can see which angle is larger by checking which one has arms that are further apart.

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

Ans: ∠XOY > ∠AOB because ∠XOY appears more wider than ∠AOB.

Page 29

Q1: How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?

Ans: There are four right angles in the window of our classroom. Yes, we also see right angles in tables, chairs, cupboards, doors, whiteboards etc. of our classroom.

Explanation: A right angle is 90°, and it is found in the corners of rectangles and squares. So, if each window has four corners, it has four right angles. Other right angles can be found in corners of tables, books, doors, etc.

Q2: 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?

Ans:

Explanation: A straight angle is 180°, which is the angle formed by a straight line. If you connect A to points directly in line with each other, you form a straight angle.

Q3: 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?

Ans: 

Explanation: A right angle is formed when two lines meet and create a 90-degree angle. In the grid, you can use the dots to guide your lines to ensure they are straight and meet at right angles.

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:

Crease 2 is perpendicular to crease 1.
a. We will get four right angles because on folding the a. paper to create a crease that is perpendicular to the first crease, the two creases intersect at a right angle, dividing the plane into four angles of 90º each.
b. 1. Take a piece of paper and fold it diagonally from one corner to opposite corner, thereby making a firm crease.
2. Now fold the paper again in such a way that the second crease becomes perpendicular to the first one.
3. Thus, we will have two creases intersecting at 90º angIe, forming four right angles.

Explanation: a. When you fold the paper to create a crease that is perpendicular to the first, the two creases intersect at a right angle, dividing the plane into four right angles of 90 degrees each.
b. The second fold must be made carefully, aligning the first crease with the edge of the paper to ensure the two creases are perpendicular. This guarantees that the angles formed are exactly 90 degrees.

Page 31

Q1: Identify acute, right, obtuse, and straight angles in the previous figures.
Ans: 

Explanation: To identify these angles, look at how much the angle opens. An acute angle is small, a right angle forms an “L” shape, an obtuse angle is larger, and a straight angle is flat.

Q2: Make a few acute angles and a few obtuse angles. Draw them in different orientations.
Ans:

Page 32

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: 

• The word acute means sharp and is used to describe angle which is less than 900 thereby appearing sharp, narrow and pointed. 
• The word obtuse means blunt and is used to describe angle which is greater than 900 and less than 1800 thereby appearing wider and less sharp.

Explanation: These terms help to visualize the nature of the angles, where an acute angle looks more pointed, and an obtuse angle looks wider and less pointed.

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:

Explanation: As the figures increase in complexity, the number of acute angles increases in a pattern. The next figure would likely have more smaller triangles, and therefore more acute angles. The pattern shows that the number of acute angles triples with each step.

Page 40

Q1: Find the degree measures of the following angles using your protractor.

Ans: 

Q2: Find the degree measures of different angles in your classroom using our protractor.
Ans: Students should try to solve this question on their own, but here’s a hint to help them along the way: Measure each angle using your protractor. If your paper protractor was made correctly, it should give you the same results as a standard protractor.

Page 41

Q3: Find the degree measures for the angles given below. Check if your paper protractor can be used here!
Ans:

Explanation: To accurately measure these angles, ensure your paper protractor is precise and that it can measure the angles in question. The larger angle will be obtuse, and the smaller one will be acute.

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

Ans: (i) Measure the smaller angle with the help of protractor.
(ii) On subtracting the smaller angle from 360º, we get the value of the required bigger angle.
Smaller angle = 100º
Required bigger angle = 360º - 100º = 260º

Explanation: The angle you are measuring appears to be an obtuse angle (greater than 90°). By following these steps, you can accurately determine the angle's degree.

Q5: Measure and write the degree measures for each of the following angles:

Ans: 

Page 42

Q6: Find the degree measures of ∠BXE, ∠CXE, ∠AXB, and ∠BXC.

Ans: ∠BXE = 115º
∠CXE = 85º
∠AXB = 65º
∠BXC = ∠AXC - ∠AXB

Q7: Find the degree measures of ∠PQR, ∠PQS, and ∠PQT.

Ans: ∠PQR = 45º
∠PQS = 80º
∠ PQT = 130º

Explanation: Each angle is measured by placing the protractor's center at the vertex Q and aligning one of the rays with the 0° mark on the protractor. The other ray's intersection with the protractor scale gives the degree measure. The sum of ∠PQR and ∠PQS should give you ∠PQT, as angles on a straight line add up.

Page 43

Q8: 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.

Ans: Students should try to solve this question on their own, but here’s a hint to help them along the way:

1. Follow the steps in the images to fold the paper into the shape shown.
2. After completing the folding steps, carefully unfold the paper to reveal the creases.
3. Draw lines along the creases with a pencil.
4. Use a protractor to measure the angles formed at the intersections of the creases.

Q9: 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!

Ans: ∠ACB = 70º ; ∠CAB = 45º ; ∠ABC= 65º
Sum of all angles of ΔABC = ∠ACB + ∠CAB + ∠ABC
= 95º + 45º + 50º
= 180º
(b) ∠ACB = 62º ; ∠CAB = 55º ; ∠ABC= 63º
Sum of all angles of ΔABC = ∠ACB + ∠CAB + ∠ABC
= 62º + 55º + 63º
= 180º
(c) ∠ACB = 97º ; ∠CAB = 30º ; ∠ABC = 53º
Sum of all angles of ΔABC = ∠ACB + ∠CAB + ∠ABC
= 97º + 30º + 53º
= 180º
We have conjectured from the given activity that the sum of all the angles of any given triangle is always 180º.

Page 45

Q1: 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.

Ans: a. A clock is divided into 12 hours and has a total of 360º.
Each hour = 360º/12 = 30º.
Therefore, at 1 o'clock the hour hand is at 1 and minute hand is at 12, thereby making an angle of 30º.
b. Angle at 2 0'clock = 2 x  30º
= 60º
Angle at 4 0'clock = 4 x 30º
= 120º
Angle at 6 0'clock = 6 x 30º
= 180º
c. Angle at 3 0'clock = 3 x 30º
= 90º
Angle at 5 0'clock = 5 x 30º
= 15º
Angle at 7 0'clock = 7 x 30º
= 210º
Angle at 8 0'clock = 8 x 30º
= 240º
Angle at 9 0'clock = 9 x 30º
= 270º
Angle at 10 0'clock = 10 x 30º
= 300º
Angle at 11 0'clock = 11 x 30º
= 330º
Angle at 12 0'clock = 12 x 30º
= 360º

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

Q3: 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?
Ans: 
Yes, we can see the angle and the angle is between the rope and the branch of the tree.

Page 46

Q4: 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?
Ans:
Yes, angles can be used to describe the slopes of the slabs. The slanting slab and invisible line perpendicular to the sides of the toy are two arms of each angle.
The slanting slab is visible and the line perpendicular to the sides of the toy are invisible.

Q5: 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.
Ans: Yes, angles can be used to describe the amount of the rotation by observing the initial and the final position of the insect.
The horizontal line and the insect itself will be the arms of the angle. The vertex will be the back end of the insect.

Page 49

Q1: 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.
Ans: All angles are ∠PAC, ∠ACL, ∠APL, ∠CLP, ∠RPL, ∠PLS, ∠LSR, ∠PRS, ∠BRS.
The guessed measurements were quite close to the actual measurements.

Page 50

Q2: Use a protractor to draw angles having the following degree measures:
a. 110º  
b. 40º 
c. 75º 
d. 112º 
e. 134º
Ans: 

Q3: 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.
Ans: The measurement of the given angle is 116º and it can be drawn by the following steps:
(i) Draw .
(ii) Place the centre point of the protractor on O and align  to the zero line.
(iii) Now starting from 0º count up to 116º and mark a point B at the label 116º.
(iv) Using a ruler join the point O and B.
∠AOB = 116º is the required angle.

Page 51

Q1: 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.
Ans:
a. An acute angle
b. An obtuse angle
c. A reflex angle

Q2: 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

Ans: 
a. ∠PTR = 30º
b. ∠PTQ = 60º
c. ∠PTW = 104º
d. ∠WTP = 360º - ∠PTW
= 360º - 104º
= 256º.

Page 53

Q1: Draw angles with the following degree measures:
a. 140°
b. 82°
c. 195°
d. 70°
e. 35°
Ans: 

Q2: Estimate the size of each angle and then measure it with a protractor:

Ans: 

Q3: Make any figure with three acute angles, one right angle, and two obtuse angles.
Ans: 
3 Acute angles = ∠COD, ∠BOC, ∠BOD
1 Right angle = ∠AOB
2 Obtuse angles = ∠AOC, ∠AOD.

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

Explanation: Each side of the ‘M’ should form a 40° angle with the base, and the central angle at the top should be 60°.

Q5: Draw the letter ‘Y’ such that the three angles formed are 150°, 60°, and 150°.
Ans: 

Explanation: The angles at the top should be wide (150°) while the central angle at the base should be narrower (60°).

Page 54

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: The Ashoka Chakra has 24 spokes and has a total of 360°.
Angle between two spokes = 360°/24 = 15°
The possible angles between two spokes are 15°, 30°, 45°, 60°, 75°.
Therefore, the largest acute angle formed between two spokes is 75°.

Explanation: Acute angles are less than 90°, and since the angle between each spoke is 15°, it is the largest acute angle.

Q7: 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. What are the possibilities for my measure?
Ans: The possibilities for your measure are 19°, 20°, 21°, 22°. 

Explanation: An acute angle is less than 90°, so doubling, tripling, or quadrupling these values still results in an acute angle. However, multiplying by 5 gives 50°, 75°, and 90°, which makes 90° an obtuse angle in some cases.