Lines and Angles — Questions and Answers
Welcome to HSLC Guru. This page gives complete, worked answers to every Work it Out exercise (2.1 to 2.7) of ASSEB Class 6 New Mathematics Chapter 2, Lines and Angles (ৰেখা আৰু কোণ), along with a set of extra practice questions.
Summary
This chapter introduces the basic ideas of geometry — the point, the line segment, the line and the ray. A point only indicates a location and has no length, width or height. The shortest path between two points is a line segment; extending it endlessly in both directions gives a line, and extending it in only one direction gives a ray.
Two rays with a common initial point form an angle. The common point is the vertex and the two rays are the arms. The measure of an angle does not depend on the length of its arms; it depends only on the amount of rotation between the two arms. A full rotation is divided into 360 equal parts, and each part is one degree (1°).
According to the amount of rotation, angles are classified as acute (0°–90°), right (90°), obtuse (90°–180°), straight (180°), reflex (180°–360°) and complete (360°). A protractor is used to measure and draw angles, while angles can be compared by the superposition method or with a transparent circular tool.
Summary: This page gives complete, worked answers to every Work it Out exercise (2.1 to 2.7) of ASSEB Class 6 New Mathematics Chapter 2, Lines and Angles (ৰেখা আৰু কোণ), covering points, line segments, lines, rays and their notation, angles, vertex and arms, comparison of angles by superposition, the degree measure and the clock, the types of angles (acute, right, obtuse, straight, reflex and complete), and measuring and drawing angles with a protractor, along with extra MCQs, fill in the blanks, true or false and short answer questions.
Textbook Questions and Answers
Below are all the Work it Out questions of the chapter with full, worked answers, exercise by exercise. First, let us understand the basic ideas with a couple of figures.
Work it Out 2.1
1. Mark a point on a piece of paper. How many lines can you draw through this point? Now take another two distinct points. How many lines can be drawn through these two points?
Answer: Through a single point we can draw an unlimited (infinite) number of lines. But through two distinct fixed points, only one line can be drawn.
2. Identify the line segments in the figure and name them. Among the six marked points, find those that lie on only one line segment and those that lie on two line segments.
Answer: The line segments in the figure are PQ, QR, RS, ST and TU. Of the six points, P and U lie on only one line segment each (they are the end points), while Q, R, S and T each lie on two line segments (each is the common end point of two segments).
3. Identify and name the rays in the figure. Is A the initial point of all the rays? If not, identify the rays whose initial point is not A.
Answer: Most of the rays in the figure start at A — for example AB, AC, AD and AF. So A is not the initial point of all the rays; the figure also has a ray that starts at another point, for example the ray EB whose initial point is E (not A).
4. Draw the picture of each of the following — (a) PQ and PT meet at the point P (b) OA, OB and OC meet at the point O (c) AB and CD intersect at the point P (d) A and B are on PQ, but C is not (e) A and B are on the line l, but O is not.
Answer: Each figure is drawn as follows —
- (a) Take one common point P and draw two rays PQ and PT from it in different directions; both meet at P.
- (b) Take a point O and draw three rays OA, OB and OC from it; all three meet at O.
- (c) Draw two lines AB and CD so that they cross each other at a single point P.
- (d) Draw a line PQ, mark two points A and B on it, and mark a point C off the line.
- (e) Draw a line l, mark two points A and B on it, and mark a point O off the line.
5. Look at the adjacent picture and write the names of — (a) Five points (b) Four rays (c) Five line segments. There is only one line in the figure. How can you name it differently?
Answer: (a) Five points: O, A, B, C and E. (b) Four rays: OA, OB, OC and OE. (c) Five line segments: OA, OB, OC, OE and EA. Since E, O and A lie on one straight line, that single line can be named in different ways — EA, AE, EO, OE or OA — all of which mean the same line.
6. In the figure, OA is a ray passing through the points P, Q and R. (a) How can the ray OA be represented differently? (b) Can OA be written as AO? If not, why? (c) Do OP, OQ, OR and OA represent the same ray?
Answer: (a) Since P, Q and R lie on the ray, OA can also be written as OP, OQ or OR (the initial point O stays the same). (b) No, OA cannot be written as AO, because the initial point of the ray is O; writing AO would make A the initial point, which is a different ray pointing in the opposite direction. (c) Yes — OP, OQ, OR and OA all represent the same ray, because they all have the same initial point O and go in the same direction.
Work it Out 2.2
1. Look around you — different shapes form angles (cardboard box, chair, gate). Find the angles here and mark the vertices and the sides of the angles.
Answer: Every corner of these objects has an angle. Angles (mostly right angles) are formed at the corners of a cardboard box, where the legs meet the seat of a chair, and where the vertical and horizontal bars of a gate meet. In each case, the point where the two edges meet is the vertex, and the two edges from it are the arms.
2. Do you see the angles in the adjacent figures? Find at least two angles in each figure and mark the sides and vertices that form the angles.
Answer: Wherever two edges (or rays) meet at a common point in a figure, an angle is formed. Finding two such places in each figure, the common point is the vertex and the two edges from it are the arms of the angle.
3. Draw an angle formed by the sides PQ and PR. Mark the sides and the vertex of the angle.
Answer: Take a point P and draw two rays PQ and PR from it in different directions to get the angle ∠QPR. Here P is the vertex and PQ and PR are the two arms.
4. How many angles are there in the adjacent figure?
Answer: Since three rays OA, OB and OC start from the vertex O, there are three angles: ∠AOB, ∠BOC and ∠AOC.
5. Name the angles marked in the figure below.
Answer: The marked angles are ∠AOB, ∠BOC and ∠AOC. (While naming an angle, the vertex O is always written in the middle.)
6. Take three points A, B and C not on a straight line. (a) How many lines can you draw through any pair of these points? (b) Name the lines. (c) How many angles do the lines produce? Name them. (d) Take another point D not on any line — how many more lines can now be drawn through any pair of points? (e) With four non-collinear points, how many lines are there in all? (f) What is the total number of angles at the four vertices?
Answer:
- (a) Three points make three pairs, so three lines can be drawn.
- (b) The three lines are AB, BC and CA.
- (c) These three lines form a triangle and produce three angles at the three vertices: ∠BAC, ∠ABC and ∠BCA.
- (d) Point D joins with the other three points and gives three more lines (AD, BD, CD).
- (e) Four points make six pairs, so there are six lines in all.
- (f) At each vertex three lines meet, and three rays make 3 angles. So the total number of angles at the four vertices is 4 × 3 = 12.
Work it Out 2.3
1. A circle is divided into 1, 2, 3, 4, 5, 6, 8, 9, 10 and 12 equal parts. Find the degree value of the angle produced at the centre in each case.
Answer: The angle at the centre of a full circle is 360°. Dividing it into n equal parts, the central angle of each part = 360° ÷ n. So —
| Number of equal parts | Central angle of one part |
|---|---|
| 1 | 360° ÷ 1 = 360° |
| 2 | 360° ÷ 2 = 180° |
| 3 | 360° ÷ 3 = 120° |
| 4 | 360° ÷ 4 = 90° |
| 5 | 360° ÷ 5 = 72° |
| 6 | 360° ÷ 6 = 60° |
| 8 | 360° ÷ 8 = 45° |
| 9 | 360° ÷ 9 = 40° |
| 10 | 360° ÷ 10 = 36° |
| 12 | 360° ÷ 12 = 30° |
Work it Out 2.4
1. How many minutes apart should the hour hand and minute hand be in order to form a right angle?
Answer: Each minute mark on the clock stands for 6° at the centre. A right angle is 90°, so the gap = 90° ÷ 6° = 15 minutes.
2. How many minutes apart should the hour hand and minute hand be in order to form an acute angle?
Answer: An acute angle is less than 90°. So the gap must be less than 15 minutes (that is, between 1 and 14 minutes), which gives an angle between 6° and 84°.
3. Classify the angles of Table 2 into acute, right, obtuse, straight, reflex and complete angles.
Answer: The rotation angles in Table 2 were 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330° and 360°. They are classified as follows —
| Type of angle | Angles from Table 2 |
|---|---|
| Acute (0°–90°) | 30°, 60° |
| Right (90°) | 90° |
| Obtuse (90°–180°) | 120°, 150° |
| Straight (180°) | 180° |
| Reflex (180°–360°) | 210°, 240°, 270°, 300°, 330° |
| Complete (360°) | 360° |
(The 0° at 12 o’clock is a zero angle with no rotation, so it does not fall into any of the above types.)
Work it Out 2.5
1. Some angles a, b, c, d, e and f are given. Observe them carefully and classify them in the table.
Answer: Each angle is classified by looking at the opening between its two arms. The six given angles are one of each type, and they fall as follows —
| Acute angle | Right angle | Obtuse angle | Straight angle | Reflex angle | Complete angle |
|---|---|---|---|---|---|
| a | b | c | d | e | f |
(An angle less than 90° is acute, exactly 90° is right, between 90° and 180° is obtuse, exactly 180° is straight, between 180° and 360° is reflex, and exactly 360° is a complete angle.)
2. In a lattice of equally spaced points, join point A to other points so that (a) an acute angle (b) an obtuse angle (c) a reflex angle is formed. Mark the angle in each case.
Answer: Draw two rays from A through the lattice points so that —
- (a) Acute angle: take the two rays close together (opening less than 90°).
- (b) Obtuse angle: take the two rays so that the opening is more than 90° but less than 180°.
- (c) Reflex angle: take the outer (larger) opening of the angle so that it is more than 180°.
3. If we combine a right angle and an acute angle, what angle will be formed?
Answer: A right angle = 90° and an acute angle is less than 90°. Their sum lies between 90° and 180°, so the angle formed is an obtuse angle.
4. If we combine two right angles and an acute angle, what angle will be formed?
Answer: Two right angles = 90° + 90° = 180°, and adding an acute angle (less than 90°) gives a sum between 180° and 270°, so the angle formed is a reflex angle.
5. From the figure, decide which angle of each pair is larger and explain why — (a) ∠AOB and ∠XOY (b) ∠AOB and ∠XOB (c) ∠AOY and ∠AOX.
Answer: In the figure the rays are in the order OA, OX, OY, OB, so —
- (a) ∠AOB is larger, because ∠XOY lies completely inside ∠AOB (∠AOB = ∠AOX + ∠XOY + ∠YOB).
- (b) ∠AOB is larger, because ∠AOB = ∠AOX + ∠XOB, so it exceeds ∠XOB by ∠AOX.
- (c) ∠AOY is larger, because Y is rotated farther from A than X (∠AOY = ∠AOX + ∠XOY).
6. From the figure, decide which is bigger and why — ∠XOY or ∠AOB?
Answer: ∠AOB is bigger. Since the rays OX and OY lie between OA and OB, the angle ∠XOY is contained inside ∠AOB; so, whatever the lengths of the arms, ∠AOB has the greater rotation.
7. Look at the triangle figures and answer. (a) How many triangles are in each figure, and what pattern is formed? (b) How many acute angles are in each figure? (c) How many acute angles will be in the next figure? (d) How many acute angles in the fifth figure of the pattern?
Answer: The figures are equilateral triangles divided into smaller equilateral triangles, with 2, 3 and 4 small triangles along each side. So the number of smallest triangles is 2² = 4, 3² = 9 and 4² = 16 respectively.
- (a) Number of small triangles: Figure I has 4, Figure II has 9, Figure III has 16 — this is the pattern of square numbers (4, 9, 16, 25, 36, …).
- (b) Each small triangle is equilateral, so all three of its angles are acute 60° angles. Hence the number of acute angles = 3 × (number of small triangles) = 3 × 4 = 12, 3 × 9 = 27 and 3 × 16 = 48 respectively.
- (c) The next (fourth) figure has 5 triangles per side, so 5² = 25 small triangles and 3 × 25 = 75 acute angles.
- (d) The fifth figure has 6 triangles per side, so 6² = 36 small triangles and 3 × 36 = 108 acute angles.
Work it Out 2.6
1. Determine the values of the angles (i)–(xii) using a protractor.
Answer: To measure, place the centre (origin) of the protractor at the vertex, align one arm with the 0° line, and read the mark where the other arm crosses the scale. Measuring the figures gives approximately the following values (confirm on the printed figure in your book) —
| Figure | Approx. value | Type |
|---|---|---|
| (i) | ≈ 50° | Acute |
| (ii) | ≈ 30° | Acute |
| (iii) | ≈ 65° | Acute |
| (iv) | ≈ 90° | Right |
| (v) | ≈ 115° | Obtuse |
| (vi) | ≈ 40° | Acute |
| (vii) | ≈ 160° | Obtuse |
| (viii) | ≈ 290° | Reflex |
| (ix) | ≈ 250° | Reflex |
| (x) | ≈ 20° | Acute |
| (xi) | ≈ 45° | Acute |
| (xii) | ≈ 130° | Obtuse |
2. Draw the following angles using a protractor — (a) 25° (b) 45° (c) 70° (d) 90° (e) 120° (f) 170° (g) 180° (h) 210° (i) 240° (j) 270° (k) 310° (l) 360°.
Answer: Draw an arm OA, place the centre of the protractor at O, align OA with the 0° line, mark a point at the required value and join it to O. For angles larger than 180° (210°, 240°, 270°, 310°), first draw a straight angle (180°) and then add the remaining part (30°, 60°, 90°, 130°). These angles are of the following types —
- Acute: 25°, 45°, 70°
- Right: 90°
- Obtuse: 120°, 170°
- Straight: 180°
- Reflex: 210°, 240°, 270°, 310°
- Complete: 360°
3. Using a protractor, determine (a) ∠POQ (b) ∠POR (c) ∠POS (d) ∠POT in the figure, and classify them as acute, right, obtuse and reflex angles.
Answer: Measuring anticlockwise from the arm OP gives approximately —
| Angle | Approx. value | Type |
|---|---|---|
| ∠POQ | ≈ 40° | Acute |
| ∠POR | ≈ 90° | Right |
| ∠POS | ≈ 130° | Obtuse |
| ∠POT | ≈ 220° | Reflex |
Identify and correct the errors: A student measured some angles with a protractor (∠X = 40°, ∠Y = 85°, ∠Z = 120°, ∠U = 60°, ∠V = 70°, ∠W = 150°). Identify the mistakes.
Answer: Such mistakes usually happen for two reasons — (1) one arm of the angle is not properly aligned with the 0° line of the protractor, or (2) the value is read from the wrong scale (a protractor has two scales, one increasing from the left and one from the right). For example, if the arm is set on 0° on the right, an acute angle must be read from the inner scale; reading the outer scale by mistake gives (180° − correct value). The correct value is obtained by first placing the arm exactly on 0°, then checking whether the angle looks acute or obtuse and reading from the correct (single) scale.
Work it Out 2.7
1. From the figure, determine (a) ∠AOD (b) ∠BOC (c) ∠COD (d) ∠BOD (e) ∠BOE.
Answer: In the figure, E, O and A lie on a straight line (so ∠EOA = 180°). The two marked angles are ∠AOB = 80° and ∠BOD = 60°, with ray OC lying between OB and OD. So —
- (a) ∠AOD = ∠AOB + ∠BOD = 80° + 60° = 140°
- (b) ∠BOC = measured from the figure ≈ 30° (∠BOC and ∠COD together make ∠BOD = 60°)
- (c) ∠COD = 60° − ∠BOC ≈ 30°
- (d) ∠BOD = 60° (given in the figure)
- (e) ∠BOE = 180° − ∠AOB = 180° − 80° = 100°
2. Draw a diagram forming two acute angles, two right angles and two obtuse angles from the same vertex.
Answer: Take one vertex O and draw several rays from it so that adjacent rays make the required angles — for example rays spaced 40° and 70° apart give two acute angles, spaced 90° give two right angles, and spaced 120° and 150° give two obtuse angles. In this way all six angles can be shown from the same vertex.
3. Draw the letter ‘Y’ so that the three angles are 150°, 60° and 150°. If both 150° angles are each reduced by 10°, what is the measure of the middle angle?
Answer: The three angles meeting at a point always add up to 360° (150° + 60° + 150° = 360°). Reducing each 150° angle by 10° makes them 140° each. So the middle angle = 360° − (140° + 140°) = 360° − 280° = 80°.
4. In which letters of the English alphabet do the lines form right angles?
Answer: Among the capital letters, E, F, H, L and T have vertical and horizontal lines that meet at right angles (90°).
5. Measure the three angles of each of the three triangles with a protractor and find the sum. Do you see any similarity between the sums?
Answer: When the three angles of each triangle are measured and added, the sum comes out to about 180° every time. However different the triangle, the sum of its three angles is always 180° — this is an important property of a triangle (you will learn more about it in the next grade).
6. The Ashoka Chakra has 24 spokes. What is the degree measure of the angle between two adjacent spokes? What is the largest obtuse angle formed between any two spokes?
Answer: The 24 spokes divide the full rotation of 360° equally, so the angle between two adjacent spokes = 360° ÷ 24 = 15°. The angles between spokes are multiples of 15° (15°, 30°, 45°, …). Among these, the obtuse ones (between 90° and 180°) are 105°, 120°, 135°, 150° and 165°; so the largest obtuse angle is 165° (= 11 × 15°).
Puzzles
Puzzle 1: I am an acute angle. If you double my measure I become a right angle, if you triple it I become obtuse, and if you quadruple it I become a straight angle. What is my measure?
Answer: Quadrupling gives a straight angle (180°), so the measure = 180° ÷ 4 = 45°. Check: double = 90° (right ✓), triple = 135° (obtuse ✓), quadruple = 180° (straight ✓).
Puzzle 2: I am an acute angle. Doubling, tripling and quadrupling my measure still give acute angles, but multiplying it by five gives an obtuse angle. What are my possible measures?
Answer: Let the measure be m. For quadruple to stay acute, 4m < 90° → m < 22.5°. For five times to be obtuse, 90° < 5m < 180° → 18° < m < 36°. Both conditions together give 18° < m < 22.5°. So possible measures are 19°, 20°, 21° or 22° (for example m = 20°: 20, 40, 60, 80 are all acute, but 100° is obtuse).
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. How many lines can be drawn through a single point? (a) one (b) two (c) three (d) infinitely many
Answer: (d) infinitely many
2. How many lines can be drawn through two distinct points? (a) one (b) two (c) infinitely many (d) none
Answer: (a) one
3. The shortest path between two points is a — (a) line (b) ray (c) line segment (d) angle
Answer: (c) line segment
4. The common point of the two rays forming an angle is called the — (a) arm (b) vertex (c) end point (d) centre
Answer: (b) vertex
5. The measure of a right angle is — (a) 45° (b) 90° (c) 180° (d) 360°
Answer: (b) 90°
6. An angle greater than 90° but less than 180° is called — (a) acute (b) right (c) obtuse (d) reflex
Answer: (c) obtuse
7. An angle greater than 180° but less than 360° is called — (a) straight (b) complete (c) obtuse (d) reflex
Answer: (d) reflex
8. The instrument used to measure angles is the — (a) ruler (b) compass (c) protractor (d) set-square
Answer: (c) protractor
9. The angle formed by the hour and minute hands of a clock at 3 o’clock is — (a) 30° (b) 60° (c) 90° (d) 120°
Answer: (c) 90°
10. A full rotation is divided into how many equal parts to make one degree? (a) 60 (b) 100 (c) 180 (d) 360
Answer: (d) 360
Fill in the Blanks
- The two rays forming an angle are called its ______. (arms)
- A full rotation is divided into ______ equal parts to make one degree. (360)
- One minute mark on a clock stands for ______ at the centre. (6°)
- A ______ angle measures 180°. (straight)
- The measure of an angle does not depend on the ______ of its arms. (length)
True or False
- A line can be drawn completely. — False (a line extends endlessly in both directions)
- A ray has one initial point. — True
- A right angle measures 100°. — False (a right angle is 90°)
- The measure of an angle depends on the length of its arms. — False
- A 360° angle is called a complete angle. — True
Short Answer Questions
1. What is the difference between a line segment and a ray?
Answer: A line segment has an end point at each end and a fixed length. A ray has one initial point and extends endlessly in one direction, so its length cannot be measured.
2. On what does the measure of an angle depend?
Answer: The measure of an angle depends only on the amount of rotation between the two arms at the vertex; it does not depend on the length of the arms.
3. How is a reflex angle measured with a protractor?
Answer: An ordinary protractor measures only up to 180°. So we measure the smaller remaining angle (the part that completes 360° with the reflex angle) and subtract it from 360°; or we extend one arm to make a straight angle (180°) and add 180° to the remaining measured angle.
4. What is an angle bisector?
Answer: A line or ray that divides an angle into two equal parts is called the angle bisector of that angle. The process of dividing an angle in this way is called angle bisection.
Key Terms
| Term | Meaning |
|---|---|
| Point | A mark that indicates only a location; it has no length or width |
| Line segment | The shortest path between two points, with an end point at each end |
| Line | A straight path extending endlessly in both directions |
| Ray | A path starting at an initial point and extending endlessly in one direction |
| Angle | The figure formed by two rays with a common initial point |
| Vertex | The common point of the two rays forming an angle |
| Arm (side) | Each of the two rays that form an angle |
| Rotation | The amount by which an arm turns about the vertex |
| Degree | The unit of measuring angles; 1/360 of a full rotation |
| Protractor | The semicircular instrument used to measure and draw angles |
| Right angle | An angle of 90° |
| Straight angle | An angle of 180° |
| Obtuse angle | An angle greater than 90° but less than 180° |
| Reflex angle | An angle greater than 180° but less than 360° |
| Angle bisector | A line that divides an angle into two equal parts |