--- title: "NCERT Solutions for Class 6 Maths Chapter 2 Lines and Angles" url: https://www.swavid.com/maths/class/6/chapter/lines-and-angles/ncert-solutions dateModified: 2026-10-07T14:48:43+00:00 --- # NCERT Solutions for Class 6 Maths Chapter 2 Lines and Angles This chapter's questions cover fundamental geometric concepts such as points, lines, line segments, rays, and angles, along with their identification, classification, construction, and measurement using protractors. Additionally, students explore geometric relationships like perpendicular lines, angle bisectors, and the sum of angles in triangles. Free PDF (26 pages): https://www.swavid.com/api/seo/pdf/ncert/maths/class-6/swavid-ncert-solutions-class-6-maths-chapter-2-lines-and-angles-1b92b6ec44.pdf ## Figure it Out ### Question 1 *2 marks · Very short answer* 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? **Solution** 1. Rihan can draw a manifold or uncountable number of lines through a single given point. 2. Sheetal can draw only one unique line passing through two distinct given points. **Answer:** Rihan can draw many lines, while Sheetal can draw only one line. > Common mistake: Thinking only a limited number of lines can pass through a single point. ### Question 2 *2 marks · Very short answer* 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? **Solution** 1. The line segments in the figure are $\overline{LM}$, $\overline{MP}$, $\overline{PQ}$, and $\overline{QR}$. 2. Points L and R lie on exactly one line segment, while points M, P, and Q lie on two line segments. **Answer:** Line segments: $\overline{LM}, \overline{MP}, \overline{PQ}, \overline{QR}$; Points on one segment: L, R; Points on two segments: M, P, Q. > Common mistake: Confusing line segments with rays or lines. ### Question 3 *2 marks · Very short answer* Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays? **Solution** 1. The rays shown in the figure are $\overrightarrow{TA}$, $\overrightarrow{TB}$, $\overrightarrow{TN}$, and $\overrightarrow{NB}$. 2. No, T is the starting point for $\overrightarrow{TA}$, $\overrightarrow{TB}$, and $\overrightarrow{TN}$, but not for $\overrightarrow{NB}$. **Answer:** Rays are $\overrightarrow{TA}, \overrightarrow{TB}, \overrightarrow{TN}, \overrightarrow{NB}$; T is not the starting point of every ray. > Common mistake: Assuming the first letter mentioned in any ray is always its starting point. ### Question 4 *3 marks · Short answer* Draw a rough figure and write labels appropriately to illustrate each of the following: a. $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ meet at O. b. $\overleftrightarrow{XY}$ and $\overleftrightarrow{PQ}$ intersect at point M. c. Line $l$ contains points E and F but not point D. d. Point P lies on $\overline{AB}$. **Part a** 1. Draw two rays starting from a common point O and label them $\overrightarrow{OP}$ and $\overrightarrow{OQ}$. Answer a: $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ meeting at O. **Part b** 1. Draw two intersecting lines and label them $\overleftrightarrow{XY}$ and $\overleftrightarrow{PQ}$ with M as their point of intersection. Answer b: $\overleftrightarrow{XY}$ and $\overleftrightarrow{PQ}$ intersecting at M. **Part c** 1. Draw a line $l$, mark points E and F on it, and mark point D outside the line. Answer c: Line $l$ containing E and F, and point D outside. **Part d** 1. Draw a line segment with endpoints A and B, and mark point P anywhere on the segment between A and B. Answer d: Point P lying on $\overline{AB}$. **Answer:** Rough figures drawn with appropriate labels as requested. > Common mistake: Not labeling the points or lines correctly according to the given notation. ### Question 5 *3 marks · Short answer* In Fig. 2.6, name: a. Five points b. A line c. Four rays d. Five line segments **Part a** 1. The five points marked in the figure are D, E, O, B, and C. Answer a: D, E, O, B, C **Part b** 1. A line in the figure is represented by $\overleftrightarrow{DE}$ (or other valid combinations containing these points). Answer b: $\overleftrightarrow{DE}$ **Part c** 1. Four rays shown in the figure are $\overrightarrow{OC}$, $\overrightarrow{OB}$, $\overrightarrow{OE}$, and $\overrightarrow{OD}$. Answer c: $\overrightarrow{OC}, \overrightarrow{OB}, \overrightarrow{OE}, \overrightarrow{OD}$ **Part d** 1. Five line segments in the figure are $\overline{DE}$, $\overline{DO}$, $\overline{DB}$, $\overline{EO}$, and $\overline{EB}$. Answer d: $\overline{DE}, \overline{DO}, \overline{DB}, \overline{EO}, \overline{EB}$ **Answer:** Identified the requested points, lines, rays, and line segments from Fig. 2.6. > Common mistake: Writing ray symbols incorrectly or omitting arrows. ### Question 6 *3 marks · Short answer* 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? b. Can we write $\overrightarrow{OA}$ as $\overrightarrow{AO}$? Why or why not? **Part a** 1. Yes, ray $\overrightarrow{OA}$ can also be named as $\overrightarrow{OB}$ because O is the starting point and point B lies on the same path extending in the direction of A. Answer a: Yes, because O is the starting point and B lies on the same ray. **Part b** 1. No, we cannot write $\overrightarrow{OA}$ as $\overrightarrow{AO}$ because $\overrightarrow{OA}$ has its starting point at O and goes towards A, whereas $\overrightarrow{AO}$ has its starting point at A and goes in the opposite direction. Answer b: No, because the starting points of $\overrightarrow{OA}$ and $\overrightarrow{AO}$ are different. **Answer:** Answered whether ray names can be interchanged with explanations. > Common mistake: Treating rays like line segments where order of letters does not matter. ## Figure it Out ### Question 1 *2 marks · Very short answer* 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** 1. Yes, angles can be found in the given bicycle picture, such as $\angle BDC$. 2. The vertex of this angle is D, with rays DC and DB as its arms. **Answer:** Yes, one angle is $\angle BDC$ with vertex D and arms $\overrightarrow{DC}$ and $\overrightarrow{DB}$. > Common mistake: Confusing the arms of the angle with line segments instead of rays starting from the vertex. ### Question 2 *2 marks · Very short answer* Draw and label an angle with arms ST and SR. **Solution** 1. Draw two rays starting from a common point S and label them ST and SR. 2. Mark the angle with a small curve at the vertex S to form $\angle TSR$ or $\angle RST$. **Answer:** An angle with arms ST and SR having vertex S. > Common mistake: Drawing line segments instead of rays or failing to mark the common starting point. ### Question 3 *2 marks · Very short answer* Explain why $\angle APB$ cannot be labelled as $\angle P$. **Solution** 1. Observe that multiple angles share the same vertex P in the given figure. 2. Using only $\angle P$ creates confusion as it does not specify which angle is being referred to. **Answer:** $\angle APB$ cannot be labelled as $\angle P$ because multiple angles meet at the same vertex P, making it unclear. > Common mistake: Using a single letter for an angle vertex when more than two rays originate from it. ### Question 4 *2 marks · Very short answer* Name the angles marked in the given figure. **Solution** 1. Identify the individual adjacent angles formed at the vertex T with arms TR, TQ, and TP. 2. Name the angles using three letters with the vertex as the middle letter. **Answer:** $\angle RTQ$ and $\angle RTP$ > Common mistake: Listing only one angle and missing the larger combined angle or vice versa. ### Question 5 *3 marks · Short answer* 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. **Solution** 1. Draw three points A, B, and C that are not on the same line and connect every pair with straight lines. 2. We get three lines, which are named as line AB, line BC, and line CA. 3. Using points A, B, and C, we can name three angles: $\angle ABC$, $\angle BCA$, and $\angle CAB$. **Answer:** 3 lines (AB, BC, CA) and 3 angles ($\angle ABC$, $\angle BCA$, $\angle CAB$). > Common mistake: Forgetting to name the lines properly or missing one of the angles. ### Question 6 *3 marks · Short answer* 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. **Solution** 1. Mark four points A, B, C, D such that no three are on one line, and draw all possible lines through pairs of points. 2. We get six lines: AB, BC, CD, DA, AC, and BD. 3. Using A, B, C, and D, we can name twelve angles: $\angle BAC$, $\angle CAD$, $\angle BAD$, $\angle ADB$, $\angle BDC$, $\angle ADC$, $\angle DCA$, $\angle ACB$, $\angle DCB$, $\angle CBD$, $\angle DBA$, and $\angle CBA$. **Answer:** 6 lines and 12 angles. > Common mistake: Counting diagonal lines incorrectly or missing some angles at the vertices. ## Figure it Out ### Question 1 *Activity* 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** 1. Fold a rectangular sheet of paper and draw a line along the crease. 2. The fold creates angles with the sides of the paper, which can be compared by superimposition. 3. By making different folds, various acute and obtuse angles are formed. 4. Observation: The angles vary depending on the slant of the fold, showing which fold creates the largest and smallest angles. **Answer:** Angles formed by paper folds can be compared using superimposition to find the largest and smallest angles. ### Question 2 *3 marks · Short answer* In each case, determine which angle is greater and why. a. $\angle AOB$ or $\angle XOY$ b. $\angle AOB$ or $\angle XOB$ c. $\angle XOB$ or $\angle XOC$ Discuss with your friends on how you decided which one is greater. **Part a** 1. Observe the given figure where $\angle XOY$ is an acute angle. 2. Note that $\angle AOB = \angle AOX + \angle XOY + \angle YOB$. 3. Therefore, $\angle AOB$ is greater than $\angle XOY$. Answer a: $\angle AOB$ is greater. **Part b** 1. Compare the two angles in the figure. 2. $\angle AOB$ completely contains $\angle XOB$ along with an extra part $\angle AOX$. 3. Therefore, $\angle AOB$ is greater than $\angle XOB$. Answer b: $\angle AOB$ is greater. **Part c** 1. Examine rays $OB$ and $OC$ relative to ray $OX$. 2. In the figure, point $B$ and point $C$ lie on the same ray or position such that $\angle XOB$ is identical to $\angle XOC$. 3. Therefore, neither angle is greater as they are equal. Answer c: None; $\angle XOB = \angle XOC$. **Answer:** (a) $\angle AOB$, (b) $\angle AOB$, (c) None ($\angle XOB = \angle XOC$) > Common mistake: Confusing ray containment with angle size. ### Question 3 *3 marks · Short answer* Which angle is greater: $\angle XOY$ or $\angle AOB$? Give reasons. **Solution** 1. Observe the two overlapping angles $\angle XOY$ and $\angle AOB$ in the figure. 2. Notice that their arms and vertices are not aligned in a way that makes direct visual comparison obvious. 3. Conclusion: By just looking at the figure, we cannot determine which angle is greater without superimposition or measurement. **Answer:** Cannot say by looking; superimposition or measurement is necessary. > Common mistake: Guessing the larger angle based on the length of the drawn rays instead of the amount of opening. ## Figure it Out ### Question 1 *2 marks · Very short answer* How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom? **Solution** 1. The window frames in our classroom are rectangular in shape, and each corner forms a right angle ($90^\circ$). 2. A standard window contains four right angles, and other examples include the corners of the blackboard, doors, and notebook covers. **Answer:** Classroom windows contain four right angles at their corners. > Common mistake: Confusing right angles with acute or obtuse angles. ### Question 2 *3 marks · Short answer* 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** 1. Observe the grid in the textbook (page 30, question 2) where point A is marked. 2. A straight angle measures $180^\circ$ and is formed by a straight line passing through the vertex. 3. By joining point A to other grid points horizontally or extending a line segment through A in opposite directions along the same row, we get a straight angle. **Answer:** Join point A horizontally to the grid points on its left and right in the same straight line. > Common mistake: Choosing grid points that do not lie on the same straight line passing through A. ### Question 3 *3 marks · Short answer* 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? **Solution** 1. Observe the grid in the textbook (page 30, question 3) where point A is marked. 2. A right angle measures $90^\circ$, which is half of a straight angle ($180^\circ$). 3. We can join point A to another grid point vertically or perpendicularly relative to a horizontal reference line passing through A to form a right angle ($90^\circ$ turn). **Answer:** Join point A to a grid point vertically above or below it, or perpendicular to the existing baseline. > Common mistake: Drawing an angle that is less than or greater than $90^\circ$ by choosing non-perpendicular grid points. ### Question 4 *3 marks · Short answer* 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. **Part a** 1. Folding the paper such that the slanting crease overlaps with itself along a perpendicular fold creates four right angles around the intersection point. 2. Each angle is exactly $\frac{1}{4}$ of a complete turn, which is a right angle ($90^\circ$). Answer a: We have four right angles because each is exactly a quarter turn. **Part b** 1. Fold the paper along the first slanting crease, then make a second fold across the intersection such that the edges of the first crease match perfectly. Answer b: Fold the paper over the existing crease so that the flat edges align, creating a perpendicular crease. **Answer:** Four right angles are formed, each being $\frac{1}{4}$ of a complete turn. > Common mistake: Failing to align the edges properly while folding. ## Figure it Out ### Question 1 *2 marks · Very short answer* Identify acute, right, obtuse and straight angles in the previous figures. **Solution** 1. Angles less than a quarter turn ($90^\circ$) are acute angles. 2. Angles equal to half of a full turn ($90^\circ$) are right angles, angles equal to a half turn ($180^\circ$) are straight angles, and angles between $90^\circ$ and $180^\circ$ are obtuse angles. **Answer:** Acute, right, obtuse, and straight angles can be identified in previous figures based on their degree measures or amounts of turn. > Common mistake: Confusing acute angles with obtuse angles by looking at arm lengths instead of the amount of turn. ### Question 2 *Activity* Make a few acute angles and a few obtuse angles. Draw them in different orientations. **Solution** 1. Draw various acute angles (less than $90^\circ$) and obtuse angles (greater than $90^\circ$ and less than $180^\circ$) in different orientations on paper. **Answer:** Acute and obtuse angles drawn in different orientations. ### Question 3 *2 marks · Very short answer* 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** 1. The words acute (meaning sharp) and obtuse (meaning blunt) were chosen because acute angles have a sharper opening compared to obtuse angles, which have wider and blunter openings. **Answer:** Acute means sharp due to a narrow opening, and obtuse means blunt due to a wider opening. > Common mistake: Describing angle measurements without relating them to the descriptive terms sharp and blunt. ### Question 4 *3 marks · Short answer* 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? **Solution** 1. The number of acute angles in the given figures are 3, 12, and 21 respectively. 2. The next figure will have 30 acute angles following the arithmetic pattern starting from 3 with a common difference of 9, or represented by the formula $3 \times n + 1$ where $n$ represents the number of inner triangles. 3. The pattern in the numbers is given by $3 \times 0 + 1 = 1$, $3 \times 1 + 1 = 4$, $3 \times 2 + 1 = 7$, and so on for the inner triangles structure. **Answer:** The number of acute angles in the next figure is 30, and the pattern follows $3 \times n + 1$. > Common mistake: Miscounting the individual acute angles inside the subdivided triangles. ## Figure it Out ### Question 1 *3 marks · Short answer* Write the measures of the following angles: a. $\angle KAL$ b. $\angle WAL$ c. $\angle TAK$ **Part a** 1. Observe the position of ray KA and ray AL on the unlabelled protractor. 2. Count the number of 1 degree unit parts between KA and AL. 3. $\angle KAL = 30^\circ$ Answer a: $\angle KAL = 30^\circ$ **Part b** 1. Observe the position of ray WA and ray AL on the unlabelled protractor. 2. Count the number of unit parts between WA and AL using the medium and large sized marks. 3. $\angle WAL = 50^\circ$ Answer b: $\angle WAL = 50^\circ$ **Part c** 1. Observe the position of ray TA and ray AK on the unlabelled protractor. 2. Count the total number of unit parts between TA and AK. 3. $\angle TAK = 120^\circ$ Answer c: $\angle TAK = 120^\circ$ **Answer:** a. $\angle KAL = 30^\circ$, b. $\angle WAL = 50^\circ$, c. $\angle TAK = 120^\circ$ > Common mistake: Confusing the inner and outer markings or miscounting the 5-degree and 10-degree intervals. ## Figure it Out ### Question 1s *3 marks · Short answer* Name the different angles in the figure and write their measures. **Solution** 1. Identify the rays from the vertex $O$: $OP, OQ, OR, OS, OT, OU$ as shown in the figure. 2. Read the degree measure for each angle combination using the protractor scales. 3. List all the angles with their correct measures: $\angle POQ = 35^\circ$, $\angle POR = 95^\circ$, $\angle POS = 125^\circ$, $\angle POT = 160^\circ$, $\angle QOR = 60^\circ$, $\angle QOS = 90^\circ$, $\angle QOT = 125^\circ$, $\angle QOU = 145^\circ$, $\angle ROS = 30^\circ$, $\angle ROT = 65^\circ$, $\angle ROU = 85^\circ$, $\angle SOT = 35^\circ$, $\angle SOU = 55^\circ$, and $\angle TOU = 20^\circ$. **Answer:** $\angle POQ = 35^\circ$, $\angle POR = 95^\circ$, $\angle POS = 125^\circ$, $\angle POT = 160^\circ$, $\angle QOR = 60^\circ$, $\angle QOS = 90^\circ$, $\angle QOT = 125^\circ$, $\angle QOU = 145^\circ$, $\angle ROS = 30^\circ$, $\angle ROT = 65^\circ$, $\angle ROU = 85^\circ$, $\angle SOT = 35^\circ$, $\angle SOU = 55^\circ$, $\angle TOU = 20^\circ$ > Common mistake: Using the wrong scale (inner or outer) on the protractor while reading the angle measures. ## Figure it Out ### Question 1 *3 marks · Short answer* Find the degree measures of the following angles using your protractor. **Part (a)** 1. Align the centre of the protractor with the vertex H and the base along HI. 2. Read the measure where the ray HJ crosses the protractor scale. 3. The measure of $\angle IHJ$ is $47^\circ$. Answer (a): $47^\circ$ **Part (b)** 1. Align the protractor to measure the angle between ray HK and ray HG. 2. Count or read the degree divisions on the scale between the two rays. 3. The measure of $\angle GHK$ is $23^\circ$. Answer (b): $23^\circ$ **Part (c)** 1. Place the protractor at vertex H with the base along HI and ray HJ extending across the scale. 2. Read the obtuse angle scale for ray HJ. 3. The measure of $\angle JHI$ is $108^\circ$. Answer (c): $108^\circ$ **Answer:** The measured angles are $\angle IHJ = 47^\circ$, $\angle GHK = 23^\circ$, and $\angle JHI = 108^\circ$ as per the textbook figure. > Common mistake: Reading the wrong scale (inner vs outer) on the protractor. ### Question 2 *Activity* Find the degree measures of different angles in your classroom using your protractor. **Solution** 1. Take a standard protractor from the geometry box. 2. Locate different angles in the classroom such as window corners, book edges, and table legs. 3. Place the centre of the protractor on the vertex and measure the angle in degrees. **Answer:** Measured various angles in the classroom using a protractor. ### Question 3 *3 marks · Short answer* Find the degree measures for the angles given below. Check if your paper protractor can be used here! **Part (i)** 1. Place the protractor center on the vertex H and align one arm with the base line. 2. Read the acute angle on the protractor scale. 3. The measure is $42^\circ$. Answer (i): $42^\circ$ **Part (ii)** 1. Place the protractor center on the vertex H and align one arm with the base line. 2. Read the obtuse angle on the protractor scale. 3. The measure is $116^\circ$. Answer (ii): $116^\circ$ **Part (iii)** 1. Examine whether the paper protractor with fixed creases can measure these specific angles. 2. Since the angles do not correspond to the standard folded fractions of $180^\circ$, the paper protractor cannot be used. Answer (iii): No, paper protractor cannot work here. **Answer:** The measures are $42^\circ$ and $116^\circ$. A paper protractor cannot be used here because the arms do not match the folded crease angles. > Common mistake: Using the paper protractor for arbitrary angle measurements instead of a standard graduated protractor. ### Question 4 *3 marks · Short answer* How can you find the degree measure of the angle given below using a protractor? **Solution** 1. Observe that a full turn around a point measures $360^\circ$. 2. Measure or use the given unmarked angle portion, which is $100^\circ$. 3. Subtract the unmarked angle from the total full-turn measure to get the marked angle: $360^\circ - 100^\circ = 260^\circ$. **Answer:** $260^\circ$ > Common mistake: Forgetting that a full rotation is $360^\circ$ and mistakenly subtracting from $180^\circ$. ### Question 5 *3 marks · Short answer* Measure and write the degree measures for each of the following angles: a. b. c. d. e. f. **Part (a)** 1. Observe the position of the ray on the protractor scale for figure a. 2. The measure is $80^\circ$. Answer (a): $80^\circ$ **Part (b)** 1. Observe the position of the ray on the protractor scale for figure b. 2. The measure is $120^\circ$. Answer (b): $120^\circ$ **Part (c)** 1. Observe the position of the ray on the protractor scale for figure c. 2. The measure is $60^\circ$. Answer (c): $60^\circ$ **Part (d)** 1. Observe the position of the ray on the protractor scale for figure d. 2. The measure is $130^\circ$. Answer (d): $130^\circ$ **Part (e)** 1. Observe the position of the ray on the protractor scale for figure e. 2. The measure is $130^\circ$. Answer (e): $130^\circ$ **Part (f)** 1. Observe the position of the ray on the protractor scale for figure f. 2. The measure is $60^\circ$. Answer (f): $60^\circ$ **Answer:** The angle measures are (a) $80^\circ$, (b) $120^\circ$, (c) $60^\circ$, (d) $130^\circ$, (e) $130^\circ$, and (f) $60^\circ$. > Common mistake: Confusing the inner and outer scales on the protractor. ### Question 6 *3 marks · Short answer* Find the degree measures of $\angle BXE$, $\angle CXE$, $\angle AXB$ and $\angle BXC$. **Solution** 1. Locate the rays XE and XB on the protractor figure to find $\angle BXE = 115^\circ - 0^\circ = 115^\circ$. 2. Locate the rays XE and XC to find $\angle CXE = 115^\circ - 30^\circ = 85^\circ$ or read directly from the scale. 3. Locate the rays XA and XB to find $\angle AXB = 65^\circ$. 4. Locate the rays XB and XC to find $\angle BXC = 30^\circ$. **Answer:** $\angle BXE = 115^\circ$, $\angle CXE = 85^\circ$, $\angle AXB = 65^\circ$, $\angle BXC = 30^\circ$ > Common mistake: Subtracting the wrong scale readings when the ray does not start at $0^\circ$. ### Question 7 *3 marks · Short answer* Find the degree measures of $\angle PQR$, $\angle PQS$ and $\angle PQT$. **Part (i)** 1. Place the protractor with its centre on Q and align arm QP with the $0^\circ$ line. 2. Read the measure where arm QR passes through the scale. 3. The measure of $\angle PQR$ is $45^\circ$. Answer (i): $\angle PQR = 45^\circ$ **Part (ii)** 1. Keep the protractor aligned with arm QP at $0^\circ$. 2. Read the measure where arm QS passes through the scale. 3. The measure of $\angle PQS$ is $100^\circ$. Answer (ii): $\angle PQS = 100^\circ$ **Part (iii)** 1. Keep the protractor aligned with arm QP at $0^\circ$. 2. Read the measure where arm QT passes through the scale. 3. The measure of $\angle PQT$ is $150^\circ$. Answer (iii): $\angle PQT = 150^\circ$ **Answer:** $\angle PQR = 45^\circ$, $\angle PQS = 100^\circ$, $\angle PQT = 150^\circ$ > Common mistake: Using the wrong scale (inner vs outer scale) on the protractor. ### Question 8 *Activity* 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** 1. Follow the given step-by-step paper folding instructions to create creases. 2. Unfold the paper fully and draw straight lines along all the creases formed. 3. Measure the angles formed at the intersection points using a protractor. **Answer:** Activity completed by folding paper, drawing lines on creases, and measuring the resulting angles with a protractor. ### Question 9 *Activity* 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. **Solution** 1. Measure all three interior angles of triangle (a) using a protractor and find their sum. 2. Repeat the measurement of the three interior angles for triangles (b) and (c) and find their sums. 3. Observe that the sum of the three angles in each triangle is $180^\circ$ and make the conjecture. **Answer:** The sum of the three interior angles of any triangle is $180^\circ$. ## Figure it Out ### Question 1 *3 marks · Short answer* 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. **Part a** 1. The total angle around the centre of a clock is $360^\circ$, which is divided into $12$ equal parts by the hour marks. 2. The angle between two successive numbers is calculated as $\frac{360^\circ}{12} = 30^\circ$. 3. At 1 o'clock, the hands are at 12 and 1, covering one part, so the angle between the hands is $30^\circ$. Answer a: The angle between two successive hour marks is $\frac{360^\circ}{12} = 30^\circ$. **Part b** 1. At 2 o'clock, the hands cover 2 parts, so the angle is $2 \times 30^\circ = 60^\circ$. 2. At 4 o'clock, the hands cover 4 parts, so the angle is $4 \times 30^\circ = 120^\circ$. 3. At 6 o'clock, the hands cover 6 parts, so the angle is $6 \times 30^\circ = 180^\circ$. Answer b: Angles at 2, 4, and 6 o'clock are $60^\circ$, $120^\circ$, and $180^\circ$ respectively. **Part c** 1. At 3 o'clock, the angle is $3 \times 30^\circ = 90^\circ$. 2. At 9 o'clock, the angle is $9 \times 30^\circ = 270^\circ$. Answer c: At 3 o'clock the angle is $90^\circ$, and at 9 o'clock the angle is $270^\circ$. **Answer:** At 1 o'clock: $30^\circ$, at 2 o'clock: $60^\circ$, at 4 o'clock: $120^\circ$, at 6 o'clock: $180^\circ$. > Common mistake: Multiplying by the wrong number of hour spaces or confusing the hour and minute hands. ### Question 2 *3 marks · Short answer* 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** 1. Yes, it is possible to express the amount by which a door is opened using an angle by measuring the rotation of the door. 2. The vertex of the angle is the point where the door meets the wall (the hinge line/pivot point). 3. The arms of the angle are the edge of the open door and the line of the wall or the door's closed position. **Answer:** Yes. Vertex is the hinge point, and the arms are the edge of the door and the wall. > Common mistake: Thinking the whole door surface forms the angle instead of the two intersecting lines or rays. ### Question 3 *2 marks · Very short answer* 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** 1. The angle can be thought of as being formed between the initial resting vertical position of the swing and the position where the swing reaches its highest turning point on either side. 2. The vertex of this angle is at the fixed point where the ropes of the swing are suspended from the tree branch. **Answer:** The angle is formed between the starting vertical position and the maximum swing position, with its vertex at the suspension point on the branch. > Common mistake: Failing to identify the vertex as the suspension point. ### Question 4 *3 marks · Short answer* 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** 1. Yes, angles can be used directly to describe the slopes of the slabs; the greater the angle, the greater the slope. 2. For each angle, one arm is the horizontal base/side and the other arm is the slanting slab. 3. The vertical arm or horizontal reference line is usually an imaginary line, while the slanting edge of the slab is visible. **Answer:** Yes, angles describe slopes. One arm is the horizontal base and the other is the slab edge; the vertical/horizontal reference is imaginary while the slab edge is visible. > Common mistake: Confusing which arm represents the slope and which represents the base. ### Question 5 *3 marks · Short answer* 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? **Solution** 1. Yes, angles can be used to describe the amount of rotation by measuring the turn needed to rotate the insect from its original position to its rotated position. 2. The vertex of the angle is the pivot point or the point around which the insect is rotated. 3. The arms of the angle are formed by the reference line (such as the horizontal line touching the insects) in its initial position and its final rotated position. **Answer:** Yes, angles measure rotation. The vertex is the rotation center, and the arms are the reference line segments in the initial and final positions. > Common mistake: Forgetting to use a common reference line or point for the vertex. ## Figure it Out ### Question 1 *3 marks · Short answer* 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. **Part (a)** 1. Observe the intersecting lines and points $A$, $B$, $C$, $D$, $P$, $R$, $L$, and $S$ in the figure in the textbook (Fig. 2.23). 2. List all possible angles formed at the intersections by identifying the rays meeting at common vertices. 3. List the angles: $\angle CAP$, $\angle ACD$, $\angle APL$, $\angle DLP$, $\angle RPL$, $\angle SLP$, $\angle PRS$, $\angle LSR$, $\angle BRS$, and $\angle CLP$. Answer (a): $\angle CAP$, $\angle ACD$, $\angle APL$, $\angle DLP$, $\angle RPL$, $\angle SLP$, $\angle PRS$, $\angle LSR$, $\angle BRS$, $\angle CLP$ **Part (b)** 1. Estimate the measure of each listed angle visually before measurement. 2. Measure each angle precisely using a protractor by placing its centre on the vertex and aligning one arm with the $0^\circ$ mark. Answer (b): Record the guessed and actual measured values in a comparison table. **Part (c)** 1. Compare the guessed values with the actual measured values in the table to check the accuracy of the estimates. Answer (c): Estimates become closer to actual measures with practice. **Answer:** Angles include $\angle CAP$, $\angle ACD$, $\angle APL$, $\angle DLP$, $\angle RPL$, $\angle SLP$, $\angle PRS$, $\angle LSR$, $\angle BRS$, and $\angle CLP$. > Common mistake: Forgetting to write the vertex as the middle letter when naming angles. ### Question 2 *3 marks · Short answer* Use a protractor to draw angles having the following degree measures: a. 110° b. 40° c. 75° d. 112° e. 134° **Part (a)** 1. Draw a base ray using a ruler. 2. Place the centre of the protractor on the starting point and align the base ray with the $0^\circ$ mark. 3. Locate $110^\circ$ on the protractor scale, mark the point, and join it to the vertex to get a $110^\circ$ angle. Answer (a): $110^\circ$ angle **Part (b)** 1. Draw a base ray. 2. Align the protractor centre and $0^\circ$ line. 3. Locate $40^\circ$, mark the point, and join it to the vertex. Answer (b): $40^\circ$ angle **Part (c)** 1. Draw a base ray. 2. Align the protractor. 3. Locate $75^\circ$ (halfway between $70^\circ$ and $80^\circ$), mark the point, and join it to the vertex. Answer (c): $75^\circ$ angle **Part (d)** 1. Draw a base ray. 2. Align the protractor. 3. Locate $112^\circ$ (two small divisions past $110^\circ$), mark the point, and join it to the vertex. Answer (d): $112^\circ$ angle **Part (e)** 1. Draw a base ray. 2. Align the protractor. 3. Locate $134^\circ$ (four small divisions past $130^\circ$), mark the point, and join it to the vertex. Answer (e): $134^\circ$ angle **Answer:** Angles of $110^\circ$, $40^\circ$, $75^\circ$, $112^\circ$, and $134^\circ$ drawn using a protractor. > Common mistake: Using the wrong scale (inner versus outer) on the protractor when measuring or drawing. ### Question 3 *3 marks · Short answer* 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** 1. Measure the given angle $\angle HIJ$ (or $\angle H$) in the figure in the textbook using a protractor. 2. Draw a base ray with a starting point, say point $I$. 3. Place the centre of the protractor on point $I$, align the base ray to the $0^\circ$ mark, and mark a point at the measured degree value. 4. Use a ruler to join the starting point to the marked point to complete the copied angle with the same measure. **Answer:** An angle drawn equal in measure to the given angle with step-by-step construction. > Common mistake: Not aligning the base arm properly with the $0^\circ$ line of the protractor. ## Figure it Out ### Question 1 *3 marks · Short answer* 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. **Part a** 1. Draw a line from point A to a grid point such that the angle formed with the reference base line is less than $90^\circ$ and greater than $0^\circ$. 2. Mark the interior with a small curve to show the acute angle. Answer a: An acute angle formed by joining A to a grid point. **Part b** 1. Draw a line from point A to a grid point such that the angle formed with the reference base line is greater than $90^\circ$ and less than $180^\circ$. 2. Mark the interior with a small curve to show the obtuse angle. Answer b: An obtuse angle formed by joining A to a grid point. **Part c** 1. Draw a line from point A to a grid point such that the angle enclosing the major region is greater than $180^\circ$ and less than $360^\circ$. 2. Mark the exterior with a curve to show the reflex angle. Answer c: A reflex angle formed by joining A to a grid point. **Answer:** Acute, obtuse, and reflex angles formed by joining point A to grid points. > Common mistake: Confusing interior and exterior curves when indicating reflex angles. ### Question 2 *3 marks · Short answer* Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. a. $\angle PTR$ b. $\angle PTQ$ c. $\angle PTW$ d. $\angle WTP$ **Part a** 1. Align the protractor's centre at vertex T with one arm along TP and read the position of TR. 2. The measure is $30^\circ$, which is less than $90^\circ$. Answer a: $\angle PTR = 30^\circ$ (Acute angle) **Part b** 1. Align the protractor's centre at vertex T with one arm along TP and read the position of TQ. 2. The measure is $60^\circ$, which is less than $90^\circ$. Answer b: $\angle PTQ = 60^\circ$ (Acute angle) **Part c** 1. Align the protractor's centre at vertex T with one arm along TP and read the position of TW. 2. The measure is $102^\circ$, which is greater than $90^\circ$ and less than $180^\circ$. Answer c: $\angle PTW = 102^\circ$ (Obtuse angle) **Part d** 1. Recognise that $\angle WTP$ is the reflex angle corresponding to the remaining portion around T. 2. Subtract the interior angle or measure the major turn to get $360^\circ - 102^\circ = 258^\circ$. Answer d: $\angle WTP = 258^\circ$ (Reflex angle) **Answer:** Measures and classifications for $\angle PTR$, $\angle PTQ$, $\angle PTW$, and $\angle WTP$. > Common mistake: Using the wrong scale on the protractor (inner vs outer scale) leading to incorrect degree readings. ## Figure it Out ### Question 1 *3 marks · Short answer* Draw angles with the following degree measures: a. 140° b. 82° c. 195° d. 70° e. 35° **Part a** 1. Draw a base ray and place the centre of the protractor on the vertex. 2. Align the base with the $0^\circ$ mark and count to $140^\circ$ on the protractor to mark the point. 3. Join the vertex and the marked point to form the $140^\circ$ angle. Answer a: A $140^\circ$ angle (obtuse angle) **Part b** 1. Draw a base ray and place the centre of the protractor on the vertex. 2. Align the base with the $0^\circ$ mark and count to $82^\circ$ using the small markings. 3. Join the vertex and the marked point to form the $82^\circ$ angle. Answer b: An $82^\circ$ angle (acute angle) **Part c** 1. Recognise that $195^\circ$ is a reflex angle greater than $180^\circ$. 2. Draw a straight angle ($180^\circ$) and add an extra $15^\circ$ turn below the base line. 3. Join the vertex to the point representing the extra turn. Answer c: A $195^\circ$ angle (reflex angle) **Answer:** Angles drawn using a protractor for measures 140°, 82°, 195°, 70°, and 35°. > Common mistake: Using the wrong scale on the protractor (inner versus outer numbers) while measuring or marking. ### Question 2 *3 marks · Short answer* Estimate the size of each angle and then measure it with a protractor: a., b., c., d., e., f. Classify these angles as acute, right, obtuse or reflex angles. **Part a** 1. Place the centre of the protractor on the vertex and align one arm with the $0^\circ$ mark. 2. Read the measure of the angle on the protractor scale. 3. Classify the angle based on its measure. Answer a: Measured angle value and its correct classification. **Part b** 1. Place the centre of the protractor on the vertex and align one arm with the $0^\circ$ mark. 2. Read the measure of the angle on the protractor scale. 3. Classify the angle based on its measure. Answer b: Measured angle value and its correct classification. **Part c** 1. Place the centre of the protractor on the vertex and align one arm with the $0^\circ$ mark. 2. Read the measure of the angle on the protractor scale. 3. Classify the angle based on its measure. Answer c: Measured angle value and its correct classification. **Answer:** Angles estimated, measured with a protractor, and classified as acute, right, obtuse, or reflex. > Common mistake: Confusing obtuse and reflex angles when measuring greater openings. ### Question 3 *3 marks · Short answer* Make any figure with three acute angles, one right angle and two obtuse angles. **Solution** 1. Draw a closed polygon (such as a six-sided polygon or irregular hexagon) with multiple vertices. 2. Ensure three interior angles are acute (less than $90^\circ$). 3. Ensure one interior angle is a right angle ($90^\circ$) and two interior angles are obtuse (greater than $90^\circ$ and less than $180^\circ$). **Answer:** A figure with three acute angles, one right angle, and two obtuse angles. > Common mistake: Drawing angles without verifying their exact category as acute, right, or obtuse. ### Question 4 *3 marks · Short answer* Draw the letter ‘M’ such that the angles on the sides are 40° each and the angle in the middle is 60°. **Solution** 1. Draw the left slanting segment of the letter 'M' forming an angle of $40^\circ$ at the bottom outer side. 2. Draw the inner peak and valley such that the middle interior angle measures $60^\circ$. 3. Complete the right side of the letter 'M' so that the right outer side angle measures $40^\circ$. **Answer:** The letter 'M' drawn with side angles of $40^\circ$ each and a middle angle of $60^\circ$. > Common mistake: Incorrectly positioning the vertex when measuring the $60^\circ$ angle in the middle. ### Question 5 *3 marks · Short answer* Draw the letter ‘Y’ such that the three angles formed are 150°, 60° and 150°. **Solution** 1. Draw a vertical base line segment pointing downwards from a central junction point. 2. Draw the two upper arms diverging from the junction point. 3. Measure and ensure the two bottom angles formed with the base line are $150^\circ$ each and the top inner angle is $60^\circ$. **Answer:** The letter 'Y' drawn with angles of $150^\circ$, $60^\circ$, and $150^\circ$. > Common mistake: Drawing acute angles instead of obtuse angles for the $150^\circ$ measurements. ### Question 6 *3 marks · Short answer* 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** 1. The total angle around the centre of a circle is $360^\circ$. 2. Divide $360^\circ$ by $24$ to find the angle between two successive spokes: $\frac{360^\circ}{24} = 15^\circ$. 3. Multiply $15^\circ$ by the maximum number of adjacent gaps such that the angle remains less than $90^\circ$: $15^\circ \times 5 = 75^\circ$. **Answer:** The angle between two adjacent spokes is $15^\circ$, and the largest acute angle formed between two spokes is $75^\circ$. > Common mistake: Multiplying by the wrong number of gaps to find the largest acute angle. ### Question 7 *3 marks · Short answer* 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** 1. Let the measure of the acute angle be $x^\circ$. 2. Since multiplying by 5 gives an obtuse angle, $90 < 5x < 180$, which gives $18 < x < 36$. 3. Checking conditions for $4x$, $3x$, and $2x$ to be acute ($<90^\circ$): $4x < 90 \implies x < 22.5$. 4. Thus, the integer possibilities for the angle $x^\circ$ are $19^\circ$, $20^\circ$, $21^\circ$, and $22^\circ$. **Answer:** The possibilities for the measure are $19^\circ$, $20^\circ$, $21^\circ$, and $22^\circ$. > Common mistake: Forgetting to check that all multiples up to 4 remain acute while only the 5th multiple enters the obtuse range. ## Frequently asked questions ### How many total questions are there in NCERT Solutions for Class 6 Maths Chapter 2 Lines and Angles? This chapter is organized into twelve sets of Figure it Out exercises containing a total of 51 questions. You can find step-by-step solutions for all these questions in the free PDF available on this SwaVid page. ### What topics are covered in the Class 6 Maths Chapter 2 questions? The questions cover basic geometric terms, line segments, rays, points, drawing and labeling angles, arms, vertices, and measuring angles using protractors. Additional topics include right angles, acute and obtuse angles, perpendicular creases, the angle sum property of triangles, and real-life angles in clocks and doors. ### Which question types are considered the most challenging in this chapter? Questions involving angle inequalities, measuring multiple angles sharing a common vertex using a protractor, and finding an angle by subtracting from a full turn are often found tricky by students. To approach them, carefully read the given figure, identify the starting points or common vertices, and apply geometric properties step by step. ### How should I write my answers to secure full marks in Class 6 Maths Chapter 2? To get full marks, always start by stating the given geometric properties or definitions clearly before performing any calculations or constructions. Refer to SwaVid's free PDF solutions on this page to understand the correct format for labeling rays, naming angles, and writing step-by-step construction methods. ### Is a free PDF of NCERT Solutions for Class 6 Maths Chapter 2 available for the 2026-27 session? Yes, complete solutions based on the new NCERT book for the 2026-27 session are available right here on this SwaVid page. You can easily download or view the free PDF to help with your daily homework and exam preparation. ## Related pages - [Class 6 Maths chapters](https://www.swavid.com/maths/class/6) Solutions written by SwaVid, a personal AI tutor for Class 6 to 10 Maths and Science. Practise this chapter free: https://www.swavid.com/start/student