Class 6 › Mathematics › Ganita Prakash Lines and Angles Class 6 Mathematics · Ganita Prakash · Chapter 2
Practice all → Summary Questions & answers Practise this chapter → View all Page 1 free about 8 min read , Full kit about 55 min read Page 1 free about 8 min read , Full kit about 55 min read View all 5 amazing Marks Booster kits → This chapter builds the basic vocabulary of plane geometry. A point determines a precise location and has no length, breadth or height; it is named with a capital letter such as \(P\). A line segment is the shortest path between two end points \(A\) and \(B\), written \(\overline{AB}\). Extending a segment endlessly in both directions gives a line, and extending it in only one direction gives a ray, which has one starting point. The chapter then introduces angles: an angle is formed by two rays sharing a common end point called the vertex, and the two rays are its arms. Angles are compared and measured in degrees using a protractor, where a full turn is \(360^\circ\). Special angles are named \u2014 a right angle is \(90^\circ\), a straight angle is \(180^\circ\), and a complete angle is \(360^\circ\). Angles smaller than \(90^\circ\) are acute, those between \(90^\circ\) and \(180^\circ\) are obtuse, and those between \(180^\circ\) and \(360^\circ\) are reflex. Students learn to draw and measure angles of given sizes, building the foundation for constructing and analysing shapes in later chapters.
Point
A precise location with no length, breadth or height, named by a capital letter such as \(P\).
Line segment
The shortest path between two end points \(A\) and \(B\), denoted \(\overline{AB}\).
Ray
A part of a line that starts at one point and extends endlessly in one direction.
Angle
A figure formed by two rays sharing a common end point called the vertex.
Right angle
An angle that measures exactly \(90^\circ\); a straight angle measures \(180^\circ\).
Protractor
An instrument marked in degrees used to measure and draw angles, with a full turn of \(360^\circ\).
Q1 What is a point in geometry? Answer Q2 What is a line segment? Answer Q3 How does a ray differ from a line? Answer Q4 What forms an angle? Answer Q5 What is the measure of a right angle? Answer Q6 What is the measure of a straight angle? Answer Q7 What is an acute angle? Answer Q8 What is an obtuse angle? Answer Q9 What is a reflex angle? Answer Q10 What instrument is used to measure angles and what is its unit? Answer Q1 1. Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point? Answer Q2 1. Sheetal marked two points on a piece of paper. How many different lines can she draw that pass through both of the points? Answer Q3 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 Q4 3. Name the rays shown in Fig. 2.5. Is \(T\) the starting point of each of these rays? Answer Q5 4. Draw a rough figure and write labels appropriately to illustrate each of the following: a. \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\) meet at \(O\). Answer Q6 4. Draw a rough figure and write labels appropriately to illustrate each of the following: b. \(\overleftrightarrow{XY}\) and \(\overleftrightarrow{PQ}\) intersect at point \(M\). Answer Q7 4. Draw a rough figure and write labels appropriately to illustrate each of the following: c. Line \(l\) contains points \(E\) and \(F\) but not point \(D\). Answer Q8 4. Draw a rough figure and write labels appropriately to illustrate each of the following: d. Point \(P\) lies on \(\overline{AB}\). Answer Q9 5. In Fig. 2.6, name: a. Five points Answer Q10 5. In Fig. 2.6, name: b. A line Answer Q11 5. In Fig. 2.6, name: c. Four rays Answer Q12 5. In Fig. 2.6, name: d. Five line segments Answer Q13 6. Here is a ray \(\overrightarrow{OA}\) (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 \(\overrightarrow{OB}\) ? Why? Answer Q14 6. Here is a ray \(\overrightarrow{OA}\) (Fig. 2.7). It starts at \(O\) and passes through the point \(A\). It also passes through the point \(B\). b. Can we write \(\overrightarrow{OA}\) as \(\overrightarrow{AO}\)? Why or why not? Answer Q1 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 Q2 2. Draw and label an angle with arms \(\overrightarrow{ST}\) and \(\overrightarrow{SR}\). Answer Q3 3. Explain why \(\angle APB\) cannot be labelled as \(\angle P\). Answer Q4 4. Name the angles marked in the given figure. Answer Q5 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. Answer Q6 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 Q1 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 Q2 2. In each case, determine which angle is greater and why. a. \(\angle AOB\) or \(\angle XOY\) Answer Q3 2. In each case, determine which angle is greater and why. b. \(\angle AOB\) or \(\angle XOB\) Answer Q4 2. In each case, determine which angle is greater and why. c. \(\angle XOB\) or \(\angle XOC\) Answer Q5 3. Which angle is greater: \(\angle XOY\) or \(\angle AOB\)? Give reasons. Answer Q1 1. How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom? Answer Q2 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 Q3 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? Answer Q4 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. Answer Q5 4. Get a slanting crease on the paper. Now, try to get another crease that is perpendicular to the slanting crease. 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 Q1 1. Identify acute, right, obtuse and straight angles in the previous figures. Answer Q2 2. Make a few acute angles and a few obtuse angles. Draw them in different orientations. Answer Q3 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 Q4 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 Q1 1. Write the measures of the following angles: a. \(\angle KAL\) Answer Q2 1. Write the measures of the following angles: b. \(\angle WAL\) Answer Q3 1. Write the measures of the following angles: c. \(\angle TAK\) Answer Q1 1. Find the degree measures of the following angles using your protractor. Answer Q2 2. Find the degree measures of different angles in your classroom using your protractor. Answer Q3 3. Find the degree measures for the angles given below. Check if your paper protractor can be used here! Answer Q4 4. How can you find the degree measure of the angle given below using a protractor? Answer Q5 5. Measure and write the degree measures for each of the following angles: a. Answer Q6 5. Measure and write the degree measures for each of the following angles: b. Answer Q7 5. Measure and write the degree measures for each of the following angles: c. Answer Q8 5. Measure and write the degree measures for each of the following angles: d. Answer Q9 5. Measure and write the degree measures for each of the following angles: e. Answer Q10 5. Measure and write the degree measures for each of the following angles: f. Answer Q11 6. Find the degree measures of \(\angle BXE\), \(\angle CXE\), \(\angle AXB\) and \(\angle BXC\). Answer Q12 7. Find the degree measures of \(\angle PQR\), \(\angle PQS\) and \(\angle PQT\). Answer Q13 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 Q14 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! Answer Q1 A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected. \(\angle U = 35^\circ\) Answer Q2 A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected. \(\angle V = 80^\circ\) Answer Q3 A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected. \(\angle W = 70^\circ\) Answer Q4 A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected. \(\angle X = 150^\circ\) Answer Q5 A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected. \(\angle Y = 120^\circ\) Answer Q6 A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected. \(\angle Z = 85^\circ\) Answer Q1 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 \(30^\circ\). Why? Answer Q2 1. Angles in a clock: b. What will be the angle at 2 o’clock? And at 4 o’clock? 6 o’clock? Answer Q3 1. Angles in a clock: c. Explore other angles made by the hands of a clock. Answer Q4 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 Q5 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 Q6 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 Q7 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? Answer Q1 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 Q2 2. Use a protractor to draw angles having the following degree measures: a. \(110^\circ\) Answer Q3 2. Use a protractor to draw angles having the following degree measures: b. \(40^\circ\) Answer Q4 2. Use a protractor to draw angles having the following degree measures: c. \(75^\circ\) Answer Q5 2. Use a protractor to draw angles having the following degree measures: d. \(112^\circ\) Answer Q6 2. Use a protractor to draw angles having the following degree measures: e. \(134^\circ\) Answer Q7 3. Draw an angle whose degree measure is the same as the angle given below: \(\angle HIJ\) Also, write down the steps you followed to draw the angle. Answer Q1 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 Answer Q2 1. In each of the below grids, join \(A\) to other grid points in the figure by a straight line to get: b. An obtuse angle Answer Q3 1. In each of the below grids, join \(A\) to other grid points in the figure by a straight line to get: c. A reflex angle Answer Q4 2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. a. \(\angle PTR\) Answer Q5 2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. b. \(\angle PTQ\) Answer Q6 2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. c. \(\angle PTW\) Answer Q7 2. Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. d. \(\angle WTP\) Answer Q1 In this figure, \(\angle TER = 80^\circ\). What is the measure of \(\angle BET\)? What is the measure of \(\angle SET\)? Answer Q1 1. Draw angles with the following degree measures: a. \(140^\circ\) Answer Q2 1. Draw angles with the following degree measures: b. \(82^\circ\) Answer Q3 1. Draw angles with the following degree measures: c. \(195^\circ\) Answer Q4 1. Draw angles with the following degree measures: d. \(70^\circ\) Answer Q5 1. Draw angles with the following degree measures: e. \(35^\circ\) Answer Q6 2. Estimate the size of each angle and then measure it with a protractor: a. Answer Q7 2. Estimate the size of each angle and then measure it with a protractor: b. Answer Q8 2. Estimate the size of each angle and then measure it with a protractor: c. Answer Q9 2. Estimate the size of each angle and then measure it with a protractor: d. Answer Q10 2. Estimate the size of each angle and then measure it with a protractor: e. Answer Q11 2. Estimate the size of each angle and then measure it with a protractor: f. Answer Q12 3. Make any figure with three acute angles, one right angle and two obtuse angles. Answer Q13 4. Draw the letter ‘M’ such that the angles on the sides are \(40^\circ\) each and the angle in the middle is \(60^\circ\). Answer Q14 5. Draw the letter ‘Y’ such that the three angles formed are \(150^\circ\), \(60^\circ\) and \(150^\circ\). Answer Q15 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 Q16 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
Which of these has no length, breadth or height and only a precise location?
A An angle B A line segment C A ray D A point
Flashcards & interactive practice 
