By using Ganita Prakash Book Class 6 Solutions and Chapter 2 Lines and Angles Class 6 NCERT Solutions Question Answer, students can improve their problem-solving skills.
Class 6 Maths Chapter 2 Lines and Angles Solutions
Lines and Angles Class 6 Solutions Questions and Answers
2.1 Point 2.2 Line Segment 2.3 Line 2.4 Ray Figure it Out (Page No. 15-17)
Question 1.
Can you help Rihan and Sheetal, find their answers?
Solution:
Yes, Rihan can draw infinite lines passing through a point and Sheetal can draw only one line passing through both of the points.
Question 2.
Name the line segments in given figure. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments?
Solution:
The line segments in the given figure are \(\overline{\mathrm{LM}}, \overline{\mathrm{MP}}, \overline{\mathrm{PQ}}, \overline{\mathrm{QR}}\). Points L and R are on exactly one of the line segments and points M, P and Q are on two of the line segments.
Question 3.
Name the rays shown in given figure. Is T the starting point of each of these rays?
Solution:
Rays in the given figure are \(\overrightarrow{\mathrm{TA}}, \overrightarrow{\mathrm{TN}}, \overrightarrow{\mathrm{TB}}\) and \(\overrightarrow{\mathrm{NB}}\).
Yes, T is the starting point of each of these rays in given figure.
Question 4.
Draw a rough figure and write labels appropriately to illustrate each of the following.
(a) \(\overleftrightarrow{O P}\) and \(\overleftrightarrow{O Q}\)meet at O.
Solution:
(b) \(\overleftrightarrow{X Y}\) and \(\overleftrightarrow{P Q}\) intersect at point M.
Solution:
(c) Line L contains points E and F but not point D.
Solution:
(d) Point P lies on AB.
Solution:
Question 5.
In the figure, name
(a) five points
(b) a line
(c) four rays
(d) five line segments
Solution:
(a) Five points are D, E, O, C and B.
(b) In the given figure, a line is \(\overrightarrow{D B}\).
(c) Four rays are \(\overrightarrow{O C}, \overrightarrow{O B}, \overrightarrow{E B}\) and \(\overrightarrow{O D}\).
(d) Five line segments are \(\overline{D E}, \overline{E O}, \overline{O B}, \overline{D O}\) and \(\overline{E B}\).
Question 6.
Here, a ray \(\overrightarrow{O A}\) in given figure. It starts at 0 and passes through the point A. It also passes through the point B.
(a) Can you also name it as \(\overrightarrow{O B}\)? Why?
(b) Can we write \(\overrightarrow{O B}\) as \(\overrightarrow{A O}\)? Why or why not?
Solution:
(a) Yes, we can also name of ray \(\overrightarrow{O A}\) as ray \(\overrightarrow{O B}\) because initial point of both is same and going on endlessly in the same direction.
(b) No, we cannot not write \(\overrightarrow{O A}\) as \(\overrightarrow{A O}\) because initial point of a ray cannot be changed.
2.5 Angle Figure it Out (Page No. 19-21)
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.
Solution:
Yes,
In this figure, O is the vertex of ∠AOB.
Question 2.
Draw and label an angle with arms ST and SR.
Solution:
An angle with arms ST and SR is ∠TSR.
Question 3.
Explain why ∠APCcannot be labelled as ∠P?
Solution:
In the given figure, PB divides ∠APC in two parts and . makes two angles ∠APB and ∠CPB.
So, ∠APC is greater than ∠APB and ∠CPB.
Therefore, ∠APC cannot be labelled as ∠P.
Question 4.
Name the angles marked in the given figure.
Solution:
In the given figure, marked angles are ∠RTQ and ∠RTP.
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 and C? Write them down and mark each of them with a curve as in given figure.
Solution:
Lines are AB, BC and AC and formed angles are ∠BAC, ∠BCA and ∠CBA.
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 and 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 and D? Write them all down and mark each of them with a curve as in given figure.
Solution:
Point A, B, C and D are as follow
Possible lines are AB, BC, CD, AD, AC and BD.
Thus, there are six lines formed and angles are ∠BAC, ∠BAD, ∠ADB, ∠ADC, ∠DCA, ∠DCB, ∠ABD, ∠ABC, ∠CAD, ∠BDC, ∠ACB and ∠DBC.
Thus, there are total twelve angles formed.
2.6 Comparing Angles Figure it Out (Page No. 23)
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?
Solution:
Do yourself.
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.
Solution:
On comparing the given angles in the figure by superimposition.
(a) ∠AOB is greater because size of ∠AOB is greater than size of ∠XOY.
(b) ∠AOB is greater because size of ∠AOB is greater than size of ∠XOB.
(c) ∠XOB and ∠XOC both are equal angle because vertex O and one ray \(\overrightarrow{O X}\) are common and arm \(\overrightarrow{O B}\) and \(\overrightarrow{O C}\) are overlaping.
Question 3.
Which angle is greater: ∠XOY or ∠AOB? Give reasons.
Solution:
On comparing by superimposition, the angles ∠XOY and ∠AOB in the given figure, we get ∠XOY is greater than ∠AOB because size of ∠XOY is greater.
2.7 Making Rotating Arms 2.8 Special Types of Angles Figure it Out (Page No. 29-31)
Question 1.
How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?
Solution:
Do yourself.
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?
Solution:
Question 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?
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.
Solution:
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 does not know the process can simply follow your description to get the right angle.
Solution:
(a) You will have four right angles now.
Explanation 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° each.
(b) To create a right angle
I. First fold Start by folding the paper so that one corner meets the opposite edge, creating a slant crease.
II. Second fold Now, fold the paper again but this time align the planting crease with the edge of the paper, ensuring the fold is perpendicular to the first crease.
III. Unfold When you open the paper, you will see that the two creases intersect at a 90° angle, forming four right angles.
Explanation The second fold must be made carefully, aligning the first crease with the edge of the paper to insure the two creases are perpendicular. This guarantees that the angles formed are exactly 90°.
2.7 Making Rotating Arms 2.8 Special Types of Angles Figure it Out (Page No. 31-32)
Question 1.
Identify acute, right, obtuse and straight angles in the given figures.
Solution:
In the first group, all the angles are less than a right angle. So, these are acute angles.
In the second group, all the angles are right angles.
In the third group, all the angles are more than a right angle. So, these are obtuse angles.
Question 2.
Make a few acute angles and a few obtuse angles. Draw them in different orientations.
Solution:
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?
Solution:
The word acute and obtuse’ are used to describe angles because they reflect the visual characteristics of the angle.
- Acute angles are called sharp because they are less than 90° and appear pointed, similar to the sharp edge of a knife.
- Obtuse angles are called ‘blunt’ because they are greater than 90° and appear wider, similar to the blunt end of an object.
Explanation These words help to visualise the nature of the angles where an acute angle looks more pointed and an obtuse angle looks wider and less pointed.
Question 4.
Find out the number of acute angles in each of the figures below.
What will be the next fiure and how many acute angles will it have? Do you notice any pattern in the numbers?
Solution:
First figure There are three acute angles in the first figure.
Second figure There are 12 acute angles in second figure (each of the 4 smaller triangles has 3 acute angles)
Third figure There are 21 acute angles in the third figure (each of the 7 smaller triangles has 3 acute angles).
Explanation As the figure increase in complexity, the number of acute angles increase in a pattern triangles and therefore more acute angles. The pattern shows that the number of cute angles triples with each step.
2.9 Measuring Angles Figure it Out (Page No. 35)
Question 1.
Write the measures of the following angles
(a) ∠KAL
(b) ∠WAL
(c) ∠TAK
Solution:
(a) The number of units of 1 degree angle between
KA and AL is 30.
∴ ∠KAL = 30°
(b) The number of units of 1 degree angle between LA and AW is 50.
∴ ∠WAL =50°
(c) The number of units of 1 degree angle between KA and AT is 120.
∴ ∠TAK = 120°
2.9 Measuring Angles Figure it Out (Page No. 40-43)
Question 1.
Find the degree measures of the following angles using your protractor.
Solution:
On measuring above angles by a protractor,
we get ∠IHJ = 48°
∠GHK = 25° and ∠IHK = 110°
Question 2.
Find the degree measures of different angles in your classroom using ycur protractor.
Solution:
Do yourself.
Question 3.
Find the degree measures for the angles given below. Check if your paper protractor can be used here!
Solution:
In first figure, we get ∠IHJ =45°
In second figure, we get ∠IHJ = 115°
Explanation To accurately measure these angles, insure your paper protractor is precise and that it can measure the angles in given figure.
Question 4.
How can you find the degree measure of the angle given below using a protractor?
Solution:
To measure the given angle
I. Place the protractor Position the centre point of the protractor at the vertex of the angle.
II. Align the baseline Align one of the angles ray with the 0° mark on the protractor’s base line.
III. Measure the angle Look at where the other ray crosses the protractor’s scale and read the degree measure.
Question 5.
Measure and write the degree measures for each of the following angles:
Solution:
Measure of given angles are
(a) 80°
(b) 120°
(c) 130°
(d) 130°
Question 6.
Find the degree measures of ∠BXE, ∠CXE, ∠AXBand ∠EXC.
Solution:
In the given figure, on measuring each angle by placing the protractor at the vertex X and reading the angles formed by the rays, we get
∠BXE = 115°, ∠CXE = 85°
∠AXB = 65° and ∠BXC = 30°
Question 7.
Find the degree measures of ∠PQR, ∠PQSand ∠PQT.
Solution:
In the given figure, on measuring each angle by placing the protractor at the vertex Q and reading the angles formed by the rays, we get
∠PQR = 45°, ∠PQS = 80° and ∠PQT = 150°
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.
Solution:
Do yourself.
Question 9.
Measure all three angles of the triangle shown in figure (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 figure (b) and figure (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.
Solution:
(a) For triangle (a), on measuring angles ∠A, ∠B and ∠C using protractor,
we get ∠A = 45°, ∠B = 65° and ∠C = 70°
On adding all the three angles, we get
45° + 65° + 70° = 180°
(b) For triangle (b), on measuring angles ∠A, ∠B and ∠C using protractor, we get
∠A = 60°, ∠B = 60° and ∠C = 60°
On adding all the angles, we get
60° + 60° + 60° = 180°
(c) For triangle (c), using protractor,
we get ∠A = 35°, ∠B = 55° and ∠C = 90°
On adding all the angles, we get
35° + 55° + 90° = 180°
Now, we can say that the sum of all the angles of a triangle is 180°.
2.9 Measuring Angles Figure it Out (Page No. 45-46)
Where are the angles?
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.
Solution:
(a) The clock is divided into 12 h, so each hour mark is 30° apart (360°- 12 = 30°).
Therefore, at 1 O’clock the hour hand is at 1 and the minute hand is at 12, forming a 30° angle.
(b) At 2 O’clock, it is 60° (i.e. 30° × 2 = 60°), at 4 O’clock, it is 120° (i.e. 30° × 4 = 120°) and at 6 O’clock, it is 180° (i.e. 30° × 6 = 180°)
(c) The angle increases by 30° for each hour. Other angle includes 90° at 3 O’clock, 150° at 5 O’clock and so on. Thus, on multiplying the hour by 30°, we can find the angle at any hour.
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?
Solution:
Yes, it is possible to express the amount by which a door is opened 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.
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?
Solution:
Yes, we can see the angle and the angle is between the rope and the branch of the tree.
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?
Solution:
Yes, angles be used to describe the slope of the slabs. Used slabs are the arms of the each angle.
The slanting slab and invisible line perpendicular to the sides of the toys are two arms of each angle.
The slanting slab is visible and the line perpendicular to the sides of the toys are 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.
Solution:
Yes, the angles be used to describe the amount of rotation. The horizontal line touching the insects and othi£r side touching with the corner point on horizontal line are the arms of angle and the touching point is vertex of the angle.
2.10 Drawing Angles Figure it Out (Page No. 49-50)
Question 1.
In the given figure, 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.
Solution:
In the given figure, all possible angles are as follows
| Angles | Guessed measurement | Actual measurement |
| ∠ACD, acute angle | 80° | 74° |
| ∠CAP, obtuse angle | 120° | 107° |
| ∠APL, acute angle | 85° | 82° |
| ∠PLD, obtuse angle | 110° | 98° |
| ∠RPL, obtuse angle | 110° | 99° |
| ∠PLS, acute angle | 85° | 83° |
| ∠PRS, obtuse angle | 110° | 103° |
| ∠RSL, acute angle | 80° | 79° |
| ∠BRS, acute angle | 85° | 78° |
Question 2.
Use a protractor to draw angles having the following degree measures
(a) 110°
(b) 40°
(c) 75°
(d) 112°
(e) 134°
Solution:
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.
Solution:
First, measuring the angle in given figure, we get ∠IHJ = 115°
Now, for drawing the same as the angle i.e. 115° follow the steps given below.
Step 1 Draw a line AB.
Step 2 Using protractor from the point A measure 115° and mark it as C.
Step 3 Join AC.
Step 4 Result is ∠BAC = 115°.
2.11 Types of Angles and their Measures Figure it Out (Page No. 51-52)
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.
Solution:
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
Solution:
(a) On measuring using protractor,
we get ∠PTR =30°
This is an acute angle because it is less than 90°.
(b) Measure of ∠PTQ = 60°
This is also an acute angle.
(c) Measure of ∠PTW = 105°
This is obtuse angle because it is greater than 90° and less than 180°.
(d) Measure of ∠WTP = 360° -105° = 255°
This is a reflex angle.
2.11 Types of Angles and their Measures Figure it Out (Page No. 53)
Question 1.
In this figure, ∠TER- 80°. What is the measure of ∠BET? What is the measure of ∠SET?
Hint Observe that ∠REB is a straight angle. Hence, the degree measure of ∠REB=180° of which 80° is covered by ∠TER. A similar argument can be applied to find the measure of ∠SET.
Solution:
In the given figure, we have
∠TER = 80° and ∠SER = 90°
∵ ∠BET =180° – ∠TER
= 180° – 80° = 100° and
∠SET = ∠SER – ∠TER
= 90° – 80°
= 10°
2.11 Types of Angles and their Measures Figure it Out (Page 53-54)
Question 1.
Draw angles with the following degree measures
(a) 140°
(b) 82°
(c) 195°
(d) 70°
(d) 35°
Solution:
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.
Solution:
First, guess the size of each angle then use a protractor to measure angles.
| Estimated Angles | Actual Angles | Types of Angles |
| (a) 40° | 45° | Acute |
| (b) 155° | 165° | Obtuse |
| (c) 105° | 120° | Obtuse |
| (d) 35° | 32° | Acute |
| (e) 105° | 100° | Obtuse |
| (f) 340° | 350° | Reflex |
Question 3.
Make any figure with three acute angles, one right angle and two obtuse angles.
Solution:
The required figure is given below
In which ∠QOS, ∠QOR and ∠ROS are three acute angles, ∠POQ is one right angle and ∠POR, ∠POS are two obtuse angles.
Question 4.
Draw the letter !W such that the angles on the sides are 40° each and the angle in the middle is 60°.
Solution:
Question 5.
Draw the letter V such that the three angles formed are 150°, 60° and 150°.
Solution:
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?
Solution:
The angle between two spokes is 15° (i.e. 360° + 24) because the Ashoka Chakra is a circle and dividing 360° by the number of spokes 24 gives the angle between each pair of adjacent spokes.
The largest acute angle formed between two spokes would be between adjacent spokes, which is 15°.
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?
Solution:
The possibilities for measures are 23°, 21° and 22°.